Equivariant Coarse (Co-)Homology Theories
We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some wel...
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| description | We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given, which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 057, 62 pages
Equivariant Coarse (Co-)Homology Theories
Christopher WULFF
Mathematisches Institut, Georg–August–Universität Göttingen,
Bunsenstr. 3-5, D-37073 Göttingen, Germany
E-mail: christopher.wulff@mathematik.uni-goettingen.de
URL: https://www.uni-math.gwdg.de/cwulff/eng.html
Received October 03, 2021, in final form July 15, 2022; Published online July 26, 2022
https://doi.org/10.3842/SIGMA.2022.057
Abstract. We present an Eilenberg–Steenrod-like axiomatic framework for equivariant
coarse homology and cohomology theories. We also discuss a general construction of such
coarse theories from topological ones and the associated transgression maps. A large part of
this paper is devoted to showing how some well-established coarse (co-)homology theories,
whose equivariant versions are either already known or will be introduced in this paper, fit
into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given
which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology
theories are even invariant under these new homotopies. They also led us to a meaningful
concept of topological actions of locally compact groups on coarse spaces.
Key words: equivariant coarse homology; equivariant coarse cohomology; equivariant coarse
assembly; equivariant coarse coassembly; generalized coarse homotopies
2020 Mathematics Subject Classification: 51F30; 55N35; 46L85
Contents
1 Introduction 2
2 Coarse spaces 3
2.1 Coarse homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Group actions on coarse spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Coarse (co-)homology theories 14
3.1 Flasqueness implies homotopy invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Examples 19
4.1 Ordinary coarse homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Ordinary coarse cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 The stable Higson corona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 K-theory of the Roe algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 From topological to coarse (co-)homology theories 37
5.1 Categories of sigma-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Generalized (co-)homology theories and the Rips complex . . . . . . . . . . . . . . . . . . 40
5.3 Single space (co-)homology theories and transgression maps . . . . . . . . . . . . . . . . . 43
5.4 Reduced (co-)homology and reduced transgression . . . . . . . . . . . . . . . . . . . . . . 46
6 Further examples 48
6.1 Singular (co-)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Alexander–Spanier (co-)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Coarse K-theory and coassembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.4 Coarse K-homology and assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References 61
mailto:christopher.wulff@mathematik.uni-goettingen.de
https://www.uni-math.gwdg.de/cwulff/eng.html
https://doi.org/10.3842/SIGMA.2022.057
2 C. Wulff
1 Introduction
Coarse (co-)homology theories have already been discussed before in various forms, see for
example the axiomatic approach by Mitchener [27] (only formulated for homology) or the spectral
approach by Bunke and Engel [5, 6], see also [7] for the equivariant version.
The work by Bunke and Engel et al. is undoubtedly the most “modern” manifestation of
this topic and also the mightiest from a conceptual point of view. However, it appears to be
unnecessarily inconvenient for most practitioners of coarse geometry and coarse index theory
à la John Roe, because it utilizes the language of infinity categories, which does not belong to
the standard repertoire of that community and is usually not required in their research.1 Instead,
coarse index invariants are often constructed directly in very specific models for certain coarse
(co-)homology groups and reformulating them in the world of infinity categories would be rather
futile.
Another basic idea of the Bunke–Engel-approach is that coarse (co-)homology theories should
always be defined on the category of all bornological coarse spaces. While this is theoretically
desirable, the plain unadulterated definitions of many examples work a priori only on certain
subcategories, e.g., the one of all proper metric spaces. Additional work has to be invested to
reformulate these definitions in order to squeeze the theories into this rather rigid set-up.
Therefore, it is imperative to not completely dismiss the complementary, “service-oriented”
approach: Instead of defining coarse (co-)homology in a way which forces the users to adapt their
examples to the theory, one can also formulate the theory in a flexible way which adapts itself to
the concrete examples. An Eilenberg–Steenrod like axiomatic set-up involving a suitable notion
of admissible categories of coarse spaces seems to be perfect in this regard, and the purpose of
the present exposition is to provide suchlike.
The present paper should somewhat be understood as an advancement of parts of Mitchener’s
aforementioned work [27], but we also set the focus a bit differently and take some newer
developments into account. First of all, we generalize the axioms to the equivariant case and also
consider cohomology theories. Second, we elaborate on how equivariant coarse (co-)homology
theories can be constructed from topological (co-)homology theories for so-called σ-locally
compact spaces by a coarsification process involving Rips complexes, which is an idea that goes
back to the definition of coarse K-theory in [10] and was discussed more generally in the non-
equivariant case in [13, Section 4]. The advantage of coarse theories obtained as coarsifications is
that they are related to the topological (co-)homology groups of Higson dominated coronas via
so-called transgression maps, whose equivariant and relative versions are new in this paper. Their
importance stems from their close connection to (co-)assembly maps. And third, a strong focus
is laid on two example sections which illustrate how already established coarse (co-)homology
theories, in particular the K-theories of stable Higson coronas and Roe algebras, fit into this
setup.
Our primary motivation for expanding further on the theory – and also for our specific choice
of contents – was to lay the foundations for the author’s subsequent publication [45] about
secondary cup and cap products on coarse (co-)homology theories, which was written completely
in the framework presented here, albeit without equivariance. For this reason, we focus mainly
on the material relevant therein instead of attempting to give a complete encyclopedic account
on all the aspects of coarse (co-)homology which have been discussed in the literature so far.
Nevertheless, we believe that our way of treating this topic can be highly suitable for other
researchers in the area as well, so that we decided to write this separate discourse about it.
In this regard it should be mentioned that – although a pretty large part of this paper is
devoted to giving examples – the equivariant versions of some of them are not being elaborated
in a very sophisticated manner. In particular, the equivariance is implemented into ordinary
1Note that the review of [6] in zbMATH independently expresses this opinion.
Equivariant Coarse (Co-)Homology Theories 3
coarse cohomology and the singular cohomology only very naively, which is sufficient for the
purpose of providing examples, but probably not optimal for applications. Alexander–Spanier
(co-)homology will be touched upon only briefly without any group actions at all. Therefore,
there is probably still room for improvement.
Two interesting by-products of our research should also be mentioned. One aspect of coarse
theory which always used to be a bit awkward to work with is the notion of coarse homotopy,
because its appearance is somewhat different from that of a topological homotopy: A coarse
homotopy between two coarse maps X → Y is a map H : X × Z→ Y with certain properties
(cf. Definition 2.15) such that the two maps can be identified with two restrictions of H to
certain subspaces of the form Xρ
ρ := {(x, ρ(x)) | x ∈ X},2 where ρ is a bornological map. It is
not only inconvenient that these boundaries Xρ
ρ of the homotopy are somewhat hidden inside of
the definition, but also that the canonical coarse maps Xρ
ρ → X do not even have to be coarse
equivalences for general bornological ρ, so the inverse X → Xρ
ρ ⊂ X × Z may be non-coarse.
Furthermore, it is in general also not possible to simply concatenate two homotopies and hence
transitivity of the homotopy relation is a priori not given. These problems can be solved, and
many authors do so, by assuming that the maps ρ are not only bornological but even controlled.
However, for most applications it is not really necessary to restrict the notion of coarse homotopies
in this way.
Our attempt to alter the definition of coarse homotopies in order to resolve these inconveniences
and transform its appearance to something resembling topological homotopies more closely led
to the discovery of the broader and more flexible notion of generalized coarse homotopies
(cf. Definition 2.16). These are defined not on X×Z but on X× I, I an closed interval, equipped
with a coarse structure in which all slices X × {t} are coarsely embedded copies of X and the
intervals {x} × I can widen arbitrarily outside of bounded subsets. Many coarse (co-)homology
theories, in particular those arising from topological (co-)homology theories by a coarsification
procedure but also the K-theory of the Roe algebra, are invariant under generalized coarse
homotopies.
A nice side-effect of our generalization of coarse homotopies is that it also leads us directly
to a meaningful notion of topological actions of locally groups on coarse spaces. We give its
definition, because we believe that such topological actions might be useful in future research,
but immediately afterwards we restrict our attention to discrete groups for the development of
our equivariant (co-)homology theories.
This paper is organized as follows. We recall the relevant definitions from coarse geometry
and introduce our new notions of generalized coarse homotopies and topological group actions on
coarse spaces in Section 2. Afterwards we state our axioms of coarse (co-)homology theories and
summarize elementary properties in Section 3. Section 4 gives basic examples, which are defined
by direct construction, before we exhibit a general procedure to obtain coarse (co-)homology
theories from topologial ones via Rips complexes and introduce the so-called transgression maps in
Section 5. The final Section 6 continues the discussion of examples and explains how transgression
maps are related to (co-)assembly maps.
2 Coarse spaces
In this section we recall the basics of coarse geometry, most of which can be found in [37].
Definition 2.1. A coarse structure on a set X is a collection E of subsets of X ×X, called the
controlled sets or entourages, which contains the diagonal and is closed under the formation of
subsets, inverses, products and finite unions. That is:
2The reason for this strange notation will become apparent in Sections 2.1 and 3.1.
4 C. Wulff
1) ∆X := {(x, x) | x ∈ X} ∈ E ,
2) if E ∈ E and E′ ⊂ E, then E′ ∈ E ,
3) if E ∈ E , then E−1 := {(y, x) | (x, y) ∈ E} ∈ E (transposition),
4) if E1, E2 ∈ E , then
E1 ◦ E2 := {(x, z) | ∃y ∈ X : (x, y) ∈ E1 ∧ (y, z) ∈ E2} ∈ E
(composition),
5) if E1, E2 ∈ E , then E1 ∪ E2 ∈ E .
The coarse structure E or the coarse space (X, E)
• is called connected if each element of X ×X is contained in some entourage,
• is said to be generated by a subset S of the power set P(X × X) of X × X if it is the
smallest coarse structure containing S.
Any subset S of P(X×X) generates a unique coarse structure as follows: It is contained in
at least one coarse structure, namely P(X ×X), and the intersection of coarse structures
is again a coarse structure. Thus, the unique coarse structure generated by S is the
intersection of all coarse structures in which it is contained. Equivalently, it consists of
all subsets of sets which are produced by taking finitely many unions, compositions and
transpositions of sets in S ∪ {∆},
• is called countably generated if it is generated by a countable subset S ⊂ P(X ×X).
This is equivalent to the existence of an increasing sequence E1 ⊂ E2 ⊂ E3 ⊂ · · · of
entourages such that each entourage of X is contained in one of them. To see this, let
S = {F1, F2, F3, . . . } and then define En as the union of the finitely many entourages
G1 ◦ · · · ◦Gm with m ≤ n and G1, . . . , Gm ∈
{
∆X , F1, . . . , Fn, F
−1
1 , . . . , F−1
n
}
.
The pair (X, E) is then called a (countably generated/connected) coarse space. If there is no
ambiguity, we will usually call X a coarse space, the coarse structure being understood implicitly.
If E is a coarse structure on X and A ⊂ X is a subset, then EA := {E ∩ (A × A) | E ∈ E}
is a coarse structure on A, called the restricted coarse structure. A pair (X,A) is called a pair
of coarse spaces if X is a coarse space and A ⊂ X, where we consider A as equipped with the
restricted coarse structure.
Important examples are of course metric spaces (X, d): an entourage in the metric coarse
structure on X is a subset of X ×X that is contained in ER := {(x, y) ∈ X ×X | d(x, y) ≤ R}
for some R ≥ 0. It is countably generated by the entourages En with n ∈ N. This construction
works even if we allow the metric to take on the value ∞. Then the metric coarse structure is
connected if and only if the metric takes on only finite values. If a coarse structure comes from a
metric in this way, we call it metrizable. Roe proved the following metrizability lemma under the
additional assumption of coarse connectedness, but its proof works equally well for all coarse
spaces.
Lemma 2.2 ([37, Theorem 2.55]). A coarse space is metrizable if and only if it is countably
generated.
If A ⊂ X, then the restricted coarse structure on A is the same as the metric coarse structure
of the restricted metric.
Equivariant Coarse (Co-)Homology Theories 5
Definition 2.3. For A ⊂ X and E ∈ E we call the set
PenE(A) := E ◦A := {x ∈ X | ∃y ∈ A : (x, y) ∈ E}
the E-penumbra of A. We will call a set B ⊂ X a penumbra of A, if it is contained in
PenE(A) for some entourage E, or equivalently, if it is equal to PenE(A) for some entourage E.
If PenE(A) = X, then we say that A is E-dense in X, and A is coarsely dense in X if it is
E-dense for some entourage E.
For A = {y} we call PenE({y}) the E-ball around y ∈ X. A subset K ⊂ X is called bounded
if it is contained in some E-ball, or equivalently, if it is itself some E-ball. A family of subsets
{Ki}i∈I , Ki ⊂ X, is called uniformly bounded if the union
⋃
i∈I Ki×Ki ⊂ X×X is an entourage;
in particular, each Ki is bounded in this case.
In the example of metric spaces, the E◦
R := {(x, y) ∈ X × X | d(x, y) < R}-penumbras of
arbitrary subsets are exactly their open R-neighborhoods for each R > 0. A subset K ⊂ X is
bounded in the coarse geometric sense iff K ⊂ PenE◦
R
({y}) = BR(y) for some R > 0 and some
y ∈ X, i.e., if it is bounded in the metric sense. A family of subsets of X is uniformly bounded
iff their diameters are uniformly bounded.
Definition 2.4. A coarse space X is called locally finite or discrete if each bounded set is finite
and it is called uniformly locally finite or uniformly discrete if for each uniformly bounded family
{Ki}i∈I of subsets of X the number of points in Ki is uniformly bounded over all i ∈ I.
The space is said to have bornologically bounded geometry if it posseses a coarsely dense and
locally finite subspace, called a discretization, and it is said to have coarsely bounded geometry if
it posseses a coarsely dense and uniformly locally finite subspace, called a uniform discretization.
The word “discrete” usually refers to a topological property, and indeed it is motivated by
the relation between coarse structure and topology which is usually required if both of them are
present at the same time.
Definition 2.5. A topological coarse space is a set X equipped with both a coarse structure E
and a locally compact Hausdorff topology T such that there is an entourage E0 which is
a neighborhood of the diagonal in X×X and every bounded subset with respect to E is relatively
compact with respect to T .
The two properties in the definition ensure that in this case our coarse geometric notion
of discreteness agrees with the topological notion of discreteness. Furthermore, using the first
property and Zorn’s lemma one sees that each topological coarse space X posseses a maximal
subset X ′ ⊂ X with the property that E0 ∩ (X ′ × X ′) ⊂ ∆X , and this maximal subset is(
E0 ∪ E−1
0
)
-dense and discrete in the topological and hence also the coarse geometric sense.
Thus:
Lemma 2.6. All topological coarse spaces have bornologically bounded geometry.
An example of a coarse space without bornologically bounded geometry is the space of all
bounded functions N→ N equipped with the supremum metric. It is, of course, not a topological
coarse space, because sufficiently large balls in it are not relatively compact. Although it is
a topologically discrete metric space, it does not contain any coarsely dense coarsely discrete
subset. More precisely, it is easy to see that if X ′ is an R-dense subset of this space, then any
3R-ball must contain infinitely many points of X ′.
Let us use this opportunity to clarify the relation between metrics and coarse structures a bit
more. The following lemma is in fact obvious from the definitions and Lemmas 2.2 and 2.6.
6 C. Wulff
Lemma 2.7. A coarse structure and a topology on a set coming from a metric constitute
a topological coarse structure if and only if the metric is proper. Hence, all proper metric spaces
have bornologically bounded geometry.
As a partial converse, all countably generated locally finite coarse structures are induced by
proper metrics.
Better still, the question of simultaneous metrizability of topologies and coarse structures has
been completely answered by Wright [42]. His main theorem is as follows.
Theorem 2.8 ([42, Theorem 2.5 and its proof]). Let dt, dc be two metrics on a set X such that
the coarse structure induced by dc contains an entourage E0 which is open with respect to the
topology induced by dt. Then there exists a metric d on X producing the same topology as dt and
quasi-isometric to dc. More precisely,
∀x, y ∈ X : d(x, y)− 1 ≤ dc(x, y) < (6R+ 1)(d(x, y) + 1)
with R > sup{⌈dc(x, y)⌉ | (x, y) ∈ E0}.
Strictly speaking, Wright did not allow metrics which take the value infinity, but the more
general case follows readily: We may always assume that dt takes only finite values and then
apply his results to each coarse component separately. The fact that the same constant R can be
chosen for all coarse components implies that the individual metrics can be combined to a new
metric d on X which is quasi-isometric to dc.
As a direct consequence, [42, Corollary 2.6] then gives a necessary and sufficient condition
for the simultaneous metrizability of a coarse structure and a topology on a set X. Note that
this result also covers cases in which the coarse structure and topology only satisfy one of the
compatibility condition, that there is an open entourage, but not the other one, i.e., bounded
subsets might not be relatively compact. This corresponds to the metric not being proper.
Instead of recalling Wright’s corollary here, we prove the following similar statement about the
metrizability of our topological coarse spaces defined in Definition 2.5.
Corollary 2.9. A topological coarse space X is metrizable, meaning that there is a necessarily
proper metric inducing both the coarse structure and topology, if and only if the coarse structure
is countably generated and the topology on each coarse component is second-countable.
Proof. If both topology and coarse structure come from a proper metric, then the coarse
structure is countably generated. Furthermore, for each point x ∈ X and each n ∈ N \ {0}
let Ux,n be a finite set of 1
n -balls covering the compact subset Bn(x). Then
⋃∞
n=1 Ux,n is a countable
base for the topology on the coarse component which x is contained in.
For the other direction, we have to find metrics dt, dc as in Theorem 2.8. For dc we simply
take the metric given by Lemma 2.2.
It suffices to construct dt on each coarse component C separately and then one can choose any
common extension to the whole space. Let C+ = C ∪{∞} denote the one-point compactification
of the locally compact Hausdorff space C and choose any x ∈ C. Furthermore, denote by
E1 ⊂ E2 ⊂ E3 ⊂ · · · an increasing sequence of entourages such that each entourage of X is
contained in one of them. Therefore each compact K ⊂ C is contained in one of the relatively
compact subsets En ◦ {x} and we conclude that the sets C+ \ En ◦ {x} are a neighborhood base
for ∞. Thus, the compact Hausdorff space C+ is second-countable, too, and hence metrizable
by Urysohn’s metrization theorem. Restricting the so obtained metric to C is what we choose
as dt|C . ■
In a topological coarse space, the relatively compact subsets are exactly the finite unions of
bounded ones. Note that these might not be bounded themselves if the space is not coarsely
connected. We generalize this notion to all coarse spaces.
Equivariant Coarse (Co-)Homology Theories 7
Definition 2.10. A subset of a coarse space is called relatively compact if it is a finite union of
bounded subsets.
Before we can recall and introduce some more properties and constructions of coarse spaces,
we will first have to discuss various properties of maps between coarse spaces.
Definition 2.11. Two maps f, g : S → Y from an arbitrary set S into a coarse space Y are
called close if the set {(f(x), g(x)) | x ∈ S} is an entourage of Y . A map f : X → Y between
two coarse spaces X, Y is called
1) controlled if f × f maps entourages of X to entourages of Y ,
2) proper if the preimages of bounded subsets of Y under f are relatively compact in X or,
equivalently, if the preimages of relatively compact subsets of Y under f are relatively
compact in X,
3) coarse if it is both controlled and proper,
4) a coarse equivalence if it is a coarse map and if there exists another coarse map g : Y → X
such that both g ◦ f and f ◦ g are close to the respective identities,
5) bornological if it maps bounded subsets to bounded subsets.
It is immediate from the definitions that controlled maps and in particular coarse maps are
bornological.
If the coarse structure on X is generated by a subset S ⊂ P(X ×X), then in order for the
map f to be controlled it is sufficent to check that f × f maps all entourages in the generating
set S to entourages of Y .
The composition of two maps between coarse spaces satisfying one of the five properties in
the definition again have this property. Furthermore, closeness is an equivalence relation and if
f, g : S → Y are close, h : Y → Z is a controlled map and e : T → S is any map, then the maps
h ◦ f ◦ e and h ◦ g ◦ e are close. In particular the following categories are well defined.
Definition 2.12. The (closeness3) category of coarse spaces is the category whose objects are
coarse spaces and whose morphisms are (closeness classes of) coarse maps.
The (closeness) category of pairs of coarse spaces is the category whose objects are pairs of
coarse spaces and whose morphisms are (closeness classes of) coarse maps between the bigger
spaces which preserve the subspaces.
The first of these will be considered a subcategory of the second via the faithful functor
X 7→ (X,∅), and sometimes we also make implicit use of the full functors (X,A) → X and
(X,A)→ A when talking about naturality under coarse maps between pairs of coarse spaces.
Coarse equivalences are exactly those coarse maps whose closeness classes are isomorphisms
in the closeness category of coarse spaces. Hence we define a coarse equivalence between pairs of
coarse spaces to be a map which represents an isomorphism in the closeness category of pairs
of coarse spaces. These are exactly the coarse equivalences between the big spaces which restrict
to coarse equivalences between the subspaces.
Finally, there are two more definitions that we have to recall.
Definition 2.13. The product of two coarse spaces (X, EX), (Y, EY ) is the cartesian product of
sets X×Y equipped with the product coarse structure EX×Y which is generated by the entourages
E ×̄ F := {((x, y), (x′, y′)) ∈ (X × Y )× (X × Y ) | (x, x′) ∈ E ∧ (y, y′) ∈ F}
for all E ∈ EX and F ∈ EY .
3The strange expression “closeness category” is inspired by the similarity to the well-established notion of
“homotopy category”.
8 C. Wulff
The product of two pairs of coarse spaces (X,A) and (X,B) is (X,A)× (Y,B) := (X ×Y,A×
Y ∪X ×B) equipped with the product coarse structure on X × Y .
To make it clear, this is not the cartesian product of coarse spaces in the category theoretic
sense, because the projection maps X × Y → X and X × Y → Y are not coarse maps: they are
controlled, but not proper. It is simply the coarse structure which is usually considered on the
set X × Y .
If (X, dX) and (Y, dY ) are metric spaces, then the product coarse structure is the one induced
by any metric in the quasi-isometry class of dX + dY or, equivalently, max{dX , dY }. Also, if EX
and EY are connected, countably generated, (uniformly) locally finite or of bornologically/coarsely
bounded geometry, then their products have the respective property, too. And if X and Y are
topological coarse spaces, then so is X × Y equipped with the product coarse structure and the
product topology.
The bounded subsets of the product coarse structure are exactly those which are contained
in the cross product of a bounded subset of X and a bounded subset of Y . If f, f ′ : S → X
are close and g, g′ : T → Y are close, then the maps f × g, f ′ × g′ : S × T → X × Y are also
close with respect to the product coarse structure. If f : X → X ′ and g : Y → Y ′ are two
maps between coarse spaces which satisfy one of the five properties of Definition 2.11, then
f × g : X×Y → X ′×Y ′ has the same property, too. In particular, we can take the cross product
of morphisms in all of the four coarse categories of Definition 2.12.
Definition 2.14. A coarse space X is called flasque if there exists a coarse map ϕ : X → X
such that:
• ϕ is close to the identity map,
• for any bounded (and hence also relatively compact) subset K ⊂ X there exists NK ∈ N
such that im(ϕn) ⊂ X \K for all n ≥ NK ,
• the family {ϕn}n∈N is equicontrolled, that is, for each entourage E there is an entourage F
such that (ϕn × ϕn)(E) ⊂ F for all n ∈ N.
A pair of spaces (X,A) is called flasque if X is flasque with the map ϕ preserving the subspace A.
The prototypical example of a flasque space are the product coarse spaces X × N with
ϕ(x, k) = (x, k + 1), where N carries the canonical metric coarse structure.
2.1 Coarse homotopies
The best-known notion of coarse homotopy has been introduced in [19, Definition 11.1] under
the name “Lipschitz homotopy”. However, a serious deficiency of their original definition has
been pointed out in [1, Remark 3.18], leading essentially to the following corrected definition of
coarse homotopy.
Definition 2.15. A coarse homotopy between two coarse maps between coarse pairs f, g:
(X,A)→ (Y,B) is a map H : X × Z→ Y which maps A× Z to B such that
• there are two bornological functions ρ± : X → Z such that for all x ∈ X, n ∈ Z we have
n ≤ ρ−(x) =⇒ H(x, n) = f(x),
n ≥ ρ+(x) =⇒ H(x, n) = g(x),
• and the restriction of H to
Xρ+
ρ−
:= {(x, n) ∈ X × Z | ρ−(x) ≤ n ≤ ρ+(x)}
is a coarse map.
Equivariant Coarse (Co-)Homology Theories 9
There seems to be some disagreement in the literature whether the functions ρ± are demanded
to be only bornological maps or even controlled or coarse maps. The author is not aware of two
maps which are coarsely homotopic in the first version but not in the second or third, although
such maps might very well exist.
Working with controlled maps is, of course, more convenient. In particular, in this case
it is also possible to reparametrize the coarse homotopy such that ρ+ ≡ 0 or ρ− ≡ 0 and
hence transitivity of the coarse homotopy relation follows immediately by gluing together two
reparametrized coarse homotopies (cf. [28, Theorem 2.4]). However, the author is not aware of
any homotopy invariance result which only works in this version but not in the one with only
bornological ρ±.
This classical notion of coarse homotopy appears to be rather unsatisfactory for two related
reasons: First, there are no clear boundaries of the homotopy at which it can be evaluated, but
rather strange open ends instead. The boundaries
Xρ±
ρ±
:=
{
(x, n) ∈ X × Z | n = ρ±(x)
}
are only hidden in the definition in the existence of ρ± and they are not even coarsely equivalent
to X, if ρ± are not controlled. And second, the relation given by coarse homotopy is not a priori
transitive, so one has only an equivalence relation generated by coarse homotopy and not given
exactly by it.
These drawbacks motivated the author to the following widening of the notion of coarse
homotopy, which brings back the true feeling of homotopy known from topology. As it happens,
it closely resembles and also generalizes an even older notion of coarse homotopy between coarse
maps between proper metric spaces that was introduced in [20, Definitions 1.2 and 1.3].
Definition 2.16. Let (X, EX) be a coarse space, I = [a, b] an interval and U = {Ux}x∈X
a collection of neigborhoods of the diagonal ∆I in I × I. We define the coarse structure EU on
X × I to be the one generated by the entourages
E ×̄∆I = {((x, s), (y, s)) | (x, y) ∈ E ∧ s ∈ I} for E ∈ EX
and
EU := {((x, s), (x, t)) | x ∈ X ∧ (s, t) ∈ Ux}.
A generalized coarse homotopy between two coarse maps f, g : (X, EX)→ (Y, EY ) is a coarse
map (X × I, EU) → (Y, EY ) for some U as above which restricts to f on X × {a} and to g on
X × {b}.
If f, g : (X,A)→ (Y,B) are coarse maps between pairs of coarse spaces, then a generalized
coarse homotopy between them is a generalized coarse homotopy between the absolute coarse
maps f, g : X → Y which takes A× I to B.
In both cases we call f generalized coarsely homotopic to g.
One big advantage of generalized coarse homotopy is that it already defines an equivalence
relation. In contrast to the usual coarse homotopy, transitivity is simply obtained by gluing
together two of these homotopies.
Note that the coarse structure EU is countably generated if and only if the coarse structure EX
is countably generated.
Furthermore, the bounded subsets of (X×I, EU ) are exactly those which are contained in K×I
for some bounded K ⊂ X. To see this, assume that K ⊂ X is bounded and choose x ∈ X. Then
K×{x} is an entourage of X and compactness of I implies that there is n ∈ N such that the n-fold
10 C. Wulff
composition Un
x = Ux◦· · ·◦Ux is equal to all of I×I. Thus, K×I = ((K×{x})×̄∆I)◦En
U ◦{(x, a)}
is bounded. Conversely, if K × I is bounded, then it is contained in
(E1 ×̄∆I) ◦ EU ◦ (E2 ×̄∆I) ◦ EU ◦ · · · ◦ EU ◦ (Ek ×̄∆I) ◦ {(x, s)}
for some entourages E1, . . . , Ek and some (x, s) ∈ X × I. But then K ⊂ E1 ◦ · · · ◦ Ek ◦ {x} is
bounded.
As a direct consequence we note that if (X, EX , TX) is a topological coarse space and TX×I
denotes the product topology on X × I, then (X × I, EU , TX×I) is again a topological coarse
space. Indeed, in this case the above characterization of bounded subsets of (X × I, EU ) implies
that they are relatively compact with respect to TX×I . Furthermore, if E0 denotes an entourage
of X which is a neighborhood of the diagonal in X ×X, then
(E0 ×̄∆I) ◦ EU ◦ E−1
U ◦ (E0 ×̄∆I)
−1 =
⋃
(x,t)∈X×I
(E0 ◦ {x} × Ux ◦ {t}︸ ︷︷ ︸
neighborhood of (x,t)
)2
is an entourage in EU which is a neighborhood of the diagonal in (X × I)2.
Of course, the interval can be reparametrized arbitrarily, but it is convenient to have some
flexibility in the notation. For example, it helps us to see how coarse homotopies H : X × Z
→ X give rise to generalized coarse homotopies: We define the family U = {Ux}x∈X of open
neighborhoods of the diagonal in [−∞,∞]2 by
Ux := [−∞, ρ−(x))2 ∪
{
(x, y) ∈ R2 | d(x, y) < 1
}
∪ (ρ+(x),∞]2
and then a coarse homotopy H̃ : (X × [−∞,∞], EU )→ (Y, EY ) between f and g in the sense of
Definition 2.16 is given by the formula
H̃(x, s) :=
f(x), s = −∞,
H(x, ⌊s⌋), s ∈ R,
g(x), s = +∞.
If H maps A × Z to B, then H̃ maps A × I to B. Hence this construction also works in the
relative case. We have just shown:
Lemma 2.17. If two coarse maps between coarse spaces or pairs of coarse spaces are coarsely
homotopic, then they are also generalized coarsely homotopic.
Although generalized coarse homotopies appear to be a lot more flexible than normal coarse
homotopies, the author is unaware of an example which disproves the converse of this lemma.
Just like the classical notion of coarse homotopy, generalized coarse homotopy is also compatible
with composition of coarse maps. One of the compositions is trivial: If H : (X × I, EU )→ (Y, EY )
is a coarse homotopy between two coarse maps f, g : X → Y and if h : Y → Z is another coarse
map, then h ◦H clearly is a coarse homotopy between h ◦ f and h ◦ g. For the other composition
let e : W → X be a coarse map. Then
e× idI : (W × I, Ee∗U )→ (X × I, EU ),
where e∗U := {Ue(w)}w∈W , is a coarse map and consequently H ◦ (e× idI) is a coarse homotopy
between f ◦ e and g ◦ e.
Again, these constructions work equally well for coarse maps between pairs of coarse spaces.
We thus obtain categories:
Equivariant Coarse (Co-)Homology Theories 11
Definition 2.18. The (generalized) homotopy category of coarse spaces is the category whose
objects are coarse spaces and whose morphisms are homotopy classes of coarse maps. Analogously,
we obtain the (generalized) homotopy category of pairs of coarse spaces.
Note that the generalized homotopy categories are quotients of the homotopy categories and
these in turn are quotients of the closeness coarse categories defined in Definition 2.12, because
closeness clearly implies coarse homotopy.
2.2 Group actions on coarse spaces
In this subsection we discuss actions of groups Γ on coarse spaces X. On spaces, we always let
the group act on the right.
At the end we will almost exclusively be interested in proper isocoarse (also known as
isometric) actions of discrete groups, mostly because our explicit construction of Γ-equivariant
coarse (co-)homology theories via a Rips complex construction in Section 5.2 only works easily in
this case. Otherwise the coarse spaces do not have Γ-invariant discretizations and we would have
to work with more complicated versions of the Rips complex instead, compare Emerson–Meyer
[11, Section 2.4] for a definition in the case of topological coarse spaces. Also, the theory of Roe
algebras and localization algebras, which we will review in Sections 4.4 and 6.4, requires proper
isometric group actions by discrete groups on proper metric spaces.
However, the idea of our generalized coarse homotopies directly prompts us to propose more
generally the following definition of “topological” actions of locally compact groups on coarse
spaces. It is totally unclear whether this notion might turn out to be useful in future research
or not and we also do not claim that this definition is final. Possibly, it has to be modified for
applications and the reader shall feel free to do so. Our definition should just serve as a proof
of concept that meaningful notions of actions of locally compact topological groups on coarse
spaces not carrying a topology exist.
Definition 2.19. A topological action of a locally compact group Γ on a coarse space (X, EX) is
a group action X × Γ → X, (x, γ) 7→ xγ which is also a controlled map with respect to some
coarse structure on X × Γ defined in the following way: Given a collection U = {Ux}x∈X of
neigborhoods of the diagonal ∆Γ in Γ× Γ, we define the coarse structure EU on X × Γ to be the
one generated by the entourages
E ×̄∆L = {((x, γ), (y, γ)) | γ ∈ L ∧ (x, y) ∈ E} for L ⊂ Γ compact and E ∈ EX
and
EU := {((x, γ), (x, γ′)) | x ∈ X ∧ (γ, γ′) ∈ Ux}.
In [11, Section 2.1] and [12, Section 2.2] Emerson and Meyer also mention continuous and
coarse actions of locally compact groups on coarse spaces, but one should keep in mind that
their coarse spaces are always topological coarse spaces and continuity is meant in the usual
topological sense. Coarseness of their group actions is exactly the statement that the entourages
E ×̄∆L are mapped to entourages, and continuity of their group action implies that we can pick
an entourage E0 which is simultaneously an open neighborhood of the diagonal and then choose
a family U as in our definition such that EU is mapped into E0. Hence their continuous and
coarse group actions are examples of our topological group actions.
Of course, the converse is not true: Our topological group actions on topological coarse spaces
are in general not continuous in the topological sense, because the definition makes no use of the
topology on the space. The word “topological” just refers to the fact that the topology of the
group has been taken into account. Also, we refrain from calling our group actions “coarse”,
because this property will always be assumed implicitly.
12 C. Wulff
As a more concrete example, if a locally compact group Γ acts continuously by quasi-isometries
on a proper metric space (X, d) such that the quasi-isometry constants are bounded on each
compact subset L ⊂ Γ, then we can simply take Ux := {(γ, γ′) ∈ Γ×Γ | d(xγ, xγ′) < 1}. An even
more particular example is the action of Rn ⋊GL(R, n) on Rn.
Note that in the definition we have only demanded that the group action is a controlled map
and not a coarse map in general. Using the obvious fact that multiplication with the inverse
X × Γ → X, (x, γ) 7→ xγ−1 is also a controlled map, it is straightforward to show that the
restrictions of the multiplication maps to the subsets X × L, L ⊂ Γ compact, are coarse maps.
Lemma 2.20. The topological actions of a discrete group on a coarse space are exactly the group
actions by coarse equivalences.
Proof. One implication of the lemma is exactly the preceding statement applied to singletons
L = {γ} ⊂ Γ.
The other direction follows with the coarse structure on X×Γ induced by the collection of sets
Ux := ∆Γ, which are neighborhoods of the diagonal due to the discreteness of Γ. With this choice,
the coarse space X × Γ decomposes as a coarse disjoint union of the spaces X × {γ}, that is,
these subsets carry the subspace coarse structure and they are closed under taking E-penumbras
for arbitrary entourages E. The claim follows readily. ■
The relation of topological group actions with the notion of generalized homotopy is the
following: Having seen that multiplication with each group element is always a coarse equivalence,
it is immediate that for any two group elements in the same path component of Γ their associated
multiplication maps are generalized coarsely homotopic.
After this excursion to locally compact topological groups we specialize to the type of group
actions which will appear in our (co-)homology theories: the proper and isocoarse actions of
discrete groups by coarse equivalences.
Definition 2.21. A discrete group Γ is said to act on a coarse space (X, E) if its acts on the
underlying set X such that the action of each γ ∈ Γ is a coarse equivalence with respect to
the coarse structure E . In this case we call X a coarse Γ-space, and if in addition A ⊂ X is
Γ-invariant, then we call (X,A) a pair of coarse Γ-spaces. Furthermore, the Γ-action, the coarse
Γ-space or the pair of coarse Γ-spaces is called
• proper if the set {γ ∈ Γ | Kγ ∩K ̸= ∅} is finite for all bounded K ⊂ X,
• isocoarse if every entourage is contained in a Γ-invariant entourage. In this case we will
call X concisely an isocoarse Γ-space or (X,A) a pair of isocoarse Γ-spaces.
If Γ itself carries a coarse structure in which bounded sets are finite, e.g., if Γ is finitely
generated and we equip it with the coarse structure coming from a word metric, then properness
of the Γ-action is equivalent to properness of the map X × Γ→ X ×X, (x, γ) 7→ (xγ, x).
Isocoarse actions are the coarse geometric analog of isometric actions and are sometimes even
called isometric (cf. [11, Definition 1]). If Γ acts isometrically on a metric space X then every
entourage E of X is contained in the Γ-invariant entourage ER for some sufficiently large R ≥ 0
and hence the action is isocoarse. Conversely, if Γ acts isocoarsely on a metric space (X, d), then
d′(x, y) := sup
γ∈Γ
d(xγ, yγ)
defines a Γ-invariant metric on X which is coarsely equivalent to d.
The equivariance is also easily implemented into our two notions of coarse homotopy as follows
and one readily checks that one immediately obtains corresponding (generalized) homotopy
categories of (pairs of) (iso-)coarse Γ-spaces.
Equivariant Coarse (Co-)Homology Theories 13
Definition 2.22. A Γ-equivariant coarse homotopy between two Γ-equivariant coarse maps
(X,A) → (Y,B) is simply a coarse homotopy between them which is equivariant as a map
X × Z→ Y . Similarily, a generalized Γ-equivariant coarse homotopy between two Γ-equivariant
coarse maps (X,A) → (Y,B) is a generalized coarse homotopy which is equivariant as a map
X × I → Y and where the coarse structure EU on X × I is constructed from a Γ-equivariant
family U = {Ux}x∈X , that is, Uxγ = Ux for all x ∈ X, γ ∈ Γ.
Note that this definition is already adapted to isocoarse actions: if X is an isocoarse Γ-space,
then the last condition, Uxγ = Ux, implies that (X × I, EU ) is an isocoarse Γ-space, too. If one is
really interested in general non-isocoarse Γ-actions, then it might seem appropriate to weaken
the condition to: for all γ ∈ Γ there is n ∈ N such that for all x ∈ X we have Uxγ ⊂ Ux ◦ · · · ◦ Ux︸ ︷︷ ︸
n times
.
In this case, (X × I, EU) is a coarse Γ-space for all coarse Γ-spaces X. However, even in the
few cases, where we do consider general non-isocoarse coarse Γ-spaces we will need the stronger
condition (Lemmas 4.4 and 4.9), so we decided to use it in all cases.
For the classical notion of coarse homotopy, there is no such distinction to be made. The
coarse structure on X × Z is independent of choices and isocoarse if and only if X is isocoarse.
The same applies to the following equivariant notion of flasqueness.
Definition 2.23. A coarse Γ-space or a pair of coarse Γ-spaces is called flasque if it is flasque as
a coarse space with ϕ being Γ-equivariant, cf. Definition 2.14.
Finally we also have to discuss the compatibility of group actions with discretizations. This
will be important for the construction of Γ-equivariant coarse (co-)homology theories in Section 5.
Lemma 2.24. Every proper isocoarse Γ-space X of bornologically bounded geometry4 has a Γ-
invariant discretization.
Proof. Let X ′ ⊂ X be a locally finite E-dense subset for some symmetric entourage E containing
the diagonal, which exists because of the coarsely bounded geometry. Due to the isocoarseness
we may also assume that E is Γ-invariant. Now, Zorn’s lemma implies that there is a maximal
Γ-invariant subset X ′′ ⊂ X with the property that for all (x1, x2) ∈ E2 ∩ (X ′′ ×X ′′) we have
x1Γ = x2Γ. This subset is E
2-dense.
We claim that it is also locally finite. Let K ⊂ X ′′ be bounded. Then there is a map
f : K → PenE(K) ∩ X ′ which maps each x ∈ K to a point f(x) ∈ X ′ with (x, f(x)) ∈ E.
If f(x1) = f(x2), then (x1, x2) = (x1, f(x1))◦ (f(x2), x2) ∈ E2∩ (X ′′×X ′′) =⇒ x1Γ = x2Γ =⇒
∃γ ∈ Γ: x2 = x1γ. Because of the properness of the Γ-action and discreteness of Γ, there are for
each x1 ∈ K only finitely many γ ∈ Γ with x1γ ∈ K. Thus, preimages of points under f are all
finite. The target of f is a bounded subset of X ′, so it is finite. Hence, the domain K is also
finite. ■
Example 2.25. Consider the proper but non-isocoarse action of Z on R2 given by n.(x, y) =
(x + n, 2ny). Although R2 has coarsely bounded geometry, we cannot find a Z-invariant dis-
cretization: Assume that for some R > 0 there is a Z-invariant R-dense subset X ′. Then for
every n ∈ N the intersection of X ′ with [n, 2R+n]×
[
0, 2n+1R
]
must contain at least 2n elements
(at least one in each square [n, 2R + n] × [2Rk, 2R(k + 1)] for k = 0, . . . , 2n − 1). Therefore,
[0, 2R]2 = (−n).
(
[n, 2R+ n]×
[
0, 2n+1R
])
intersects X ′ also in at least 2n points. As n ∈ N was
arbitrary, the intersection of X ′ with [0, 2R]2 cannot be finite.
This shows that we cannot dispense with the isocoarseness assumption.
Lemma 2.26. Every proper isocoarse Γ-space X of bornologically bounded geometry has a Γ-
invariant discretization X ′ which allows a Γ-equivariant coarse equivalence π : X → X ′ which is
the identity on X ′ and hence is a coarse inverse up to closeness to the inclusion ι : X ′ ⊂ X.
4Recall that this means that X posseses a discretization, see Definition 2.4.
14 C. Wulff
Proof. Let X ′′ ⊂ X be a Γ-invariant discretization, which exists by the previous lemma. The
problem is that points of X might have stabilizers which don’t appear in X ′′ and hence we cannot
construct a Γ-equivariant coarse map X → X ′′. Instead, X ′′ has to be enlarged first.
Assume that E is a symmetric Γ-invariant entourage containing the diagonal such that X ′′
is E-dense in X. Due to the properness of the group action, for each x ∈ X the set {γ ∈ Γ |
PenE(x)γ ∩ PenE(x) ̸= ∅} is finite. All stabilizers Γy of points y ∈ PenE(x) are contained in
this set and therefore the set S(x) := {Γy | y ∈ PenE(x)} of all such stabilizers is finite.
For each Γ-orbit of O ⊂ X ′′ we choose one fixed representative xO, i.e., O = xOΓ, and
afterwards we choose for each stabilizer Σ ∈ S(xO) a representative yΣ, i.e., Σ = ΓyΣ . Let X
′ be
the union of X ′′ with all the orbits yΣΓ for all orbits O in X ′′ and all Σ ∈ S(xO). The choices
have been made in such a way that X ′ is again locally finite and therefore it is also a Γ-invariant
discretization.
Note that for each x ∈ X there is now a point x′ ∈ X ′ ∩ PenE2(x) with the same stabilizer.
This allows us to choose a Γ-equivariant coarse map X → X ′ which is the identity on X ′ and
such that the composition X → X ′ ⊂ X is E2-close to the identity. ■
3 Coarse (co-)homology theories
Not all Γ-equivariant coarse (co-)homology theories will be defined on the whole category of
pairs of coarse Γ-spaces, but only on certain sub-categories.
Definition 3.1. An admissible Γ-coarse category is a full sub-category C of the category of pairs
of coarse Γ-spaces which contains ∅ = (∅,∅) and satisfies the following condition. If (X,A) is
an object in C, then so are
(X,X), (A,A), X = (X,∅), A = (A,∅).
Examples of admissible Γ-coarse categories are those consisting of all pairs of coarse Γ-spaces
satisfying any combination of the properties isocoarseness (Definition 2.21), coarse connectedness,
countably generatedness (Definition 2.1), (uniform) discreteness, bornologically/coarsely bounded
geometry (Definition 2.4). Also, the categories of topological coarse spaces and proper metric
spaces map forgetfully onto admissible categories when forgetting the topology or the metric,
respectively.
This definition is inspired by Eilenberg–Steenrod’s notion of admissibility in [9, p. 5]. The
biggest difference is that we only consider full sub-categories and therefore do not have to put
additional conditions on the morphisms. Note that this also effects the notion of homotopy:
Eilenberg and Steenrod only consider homotopies that are morphisms in the admissible category.
In particular the domains must be objects of the admissible category. In contrast to their intrinsic
notion of homotopy we do not postulate that the domains X × Z, Xρ+
ρ− or (X × I, EU) of our
homotopies are objects in the admissible category (although they usually are) and hence we get
an extrinsic notion of homotopy.
There are some examples of sub-categories with smaller morphism sets on which (co-)homo-
logy theories can also be defined meaningfully. For example, one could consider the category of
topological coarse spaces together with all continuous coarse maps or the category of all coarse
spaces together with all rough maps, that is, coarse maps f : X → Y for which preimages of
entourages of Y under f × f are entourages of X. However, it is the author’s opinion that the
corresponding (co-)homology theories should then be called “topological coarse (co-)homology”
or “rough (co-)homology”, respectively, but not “coarse (co-)homology”.
Our admissible coarse categories together with coarse homotopy or generalized coarse homotopy
and the following notion of excisions fit into the framework of Eilenberg–Steenrod’s h-categories
[9, Definition 9.1 in Section IV.9] except that we did not single out certain spaces as “points”.
Equivariant Coarse (Co-)Homology Theories 15
Definition 3.2. We call an inclusion map between objects of C of the form (X\C,A\C)→ (X,A)
with C ⊂ A ⊂ X an excision, if for every entourage E of X there is an entourage F of X such
that PenE(A) \ C ⊂ PenF (A \ C).
This is our coarse geometric analogue of the topological excision condition that the closure
of C lies within the interior of A. It says that C sits sufficiently far inside of A such that the
difference A \ C does not change from a coarse geometric perspective if we enlarge A to an
arbitrarily large penumbra.
If we write B := X \ C, then the condition is equivalent to the coarse excisiveness condition
for the covering X = A ∪B known from [23]:
∀E ∈ EX∃F ∈ EX : PenE(A) ∩ PenE(B) ⊂ PenF (A ∩B). (3.1)
One direction is trivial. For the other we observe that
PenE(A) ∩ PenE(B) ⊂ PenE(PenE−1◦E(A) ∩B) = PenE(PenE−1◦E(A) \ C)
⊂ PenE(PenF (A \ C)) = PenE◦F (A ∩B)
for a sufficiently large entourage F .
If A,B ⊂ X are arbitrary subspaces which satisfy (3.1), then we call (X;A,B) an excisive
triad in C, even if A and B do not cover X.
We will now define coarse (co-)homology theories as (co-)homologies on these h-categories, but
without the dimension and additivity axiom, because they would only be unnecessary constraints.
Definition 3.3. Let C be an admissible category of pairs of coarse Γ-spaces.
• A Γ-equivariant coarse homology theory on C is a collection of covariant functors
{
EXΓ
p
}
p∈Z
from C to the category of Abelian groups together with a collection
{
∂Γp
}
p∈Z of natural
transformations, the connecting homomorphisms ∂Γp : EX
Γ
p (X)→ EXΓ
p−1(A) for all objects
(X,A) in C, satisfying the below-mentioned homotopy, exactness and excision axioms.
• A Γ-equivariant coarse cohomology theory on C is a collection of contravariant functors{
EXp
Γ
}
p∈Z from C to the category of Abelian groups together with a collection
{
δpΓ
}
p∈Z
of natural transformations, the connecting homomorphisms δpΓ : EX
p
Γ(A)→ EXp+1
Γ (X) for
all objects (X,A) in C, satisfying the below-mentioned homotopy, exactness and excision
axioms.
Homotopy. If two pairs of Γ-equivariant coarse maps are Γ-equivariantly coarsely homotopic,
then they induce the same maps on (co-)homology. That is, (co-)homology factors through the
homotopy category of C.
Exactness. Each pair of coarse Γ-spaces (X,A) induces a long exact sequence
· · · → EXΓ
p (A)
i∗−→ EXΓ
p (X)
j∗−→ EXΓ
p (X,A)
∂p−→ EXΓ
p−1(A)→ · · ·
or
· · · → EXp−1
Γ (A)
δp−1
−−−→ EXp
Γ(X,A)
j∗−→ EXp
Γ(X)
i∗−→ EXp
Γ(A)→ · · · ,
respectively, where (A,∅)
i−→ (X,∅)
j−→ (X,A) are the inclusion maps.
Excision. Excisions (X \C,A\C) ⊂ (X,A) induce isomorphisms EXΓ
∗ (X \C,A\C) ∼= EXΓ
∗ (X,A)
or EX∗
Γ(X,A)
∼= EX∗
Γ(X \ C,A \ C), respectively.
If Γ = 1 is the trivial group, then we call EX∗ := EXΓ
∗ simply a coarse homology theory or
EX∗ := EX∗
Γ a coarse cohomology theory, respectively.
16 C. Wulff
Standard constructions from algebraic topology (cf. [38, Chapter 4, Section 8, Theorem 5]
and [18, Section 2.3]) also show that there are a long exact sequence for triples and a coarse
Mayer–Vietoris sequence.
Proposition 3.4. Let EXΓ
∗ be a Γ-equivariant coarse homology theory or EX∗
Γ a Γ-equivariant
coarse co-homology theory.
• If X ⊃ A ⊃ B are coarse Γ-space such that (X,A), (X,B) and (A,B) are objects in C,
then there are long exact sequences
· · · → EXΓ
p (A,B)
i∗−→ EXΓ
p (X,B)
j∗−→ EXΓ
p (X,A)→ EXΓ
p−1(A,B)→ · · ·
or
· · · → EXp−1
Γ (A,B)→ EXp
Γ(X,A)
j∗−→ EXp
Γ(X,B)
i∗−→ EXp
Γ(A,B)→ · · · ,
respectively, where (A,B)
i−→ (X,B)
j−→ (X,A) are the inclusion maps and the connecting
homomorphisms are the compositions of the connecting homomorphisms of the long exact
sequence associated to the pair (X,A) with the homomorphisms induced by the inclusion
(A,∅)→ (A,B).
• Let X be a coarse Γ-space with X = A ∪ B satisfying (3.1) and such that (X,A) and
(B,A ∩B) are objects in C or (X,B) and (A,A ∩B) are objects in C. Then there are the
coarse Mayer–Vietoris sequences
· · · → EXΓ
p (A ∩B)
(i∗,j∗)−−−−→ EXΓ
p (A)⊕ EXΓ
p (B)
k∗−l∗−−−→ EXΓ
p (X)
∂MV
−−−→ EXΓ
p−1(A ∩B)→ · · ·
or
· · · → EXp−1
Γ (A ∩B)
δMV−−→ EXp
Γ(X)
(k∗,l∗)−−−−→ EXp
Γ(A)⊕ EXp
Γ(B)
i∗−j∗−−−→ EXp
Γ(A ∩B)→ · · · ,
respectively, where i : A ∩ B → A, j : A ∩ B → B, k : A → X and l : B → X denote
the inclusion maps. The connecting homomorphisms ∂MV, δMV are defined by composing
the connecting homomorphisms for the pair (B,A ∩ B) with inverse of the isomorphism
induced by the excision (B,A ∩B) ⊂ (X,A) and the homomorphism induced by inclusion
X 7→ (X,A), or analogously with the roles of A and B interchanged, depending on which
of the two possible hypotheses is true.
The axioms of Definition 3.3 are the absolute minimum needed for a meaningful notion
of (co-)homology, but there are also some additional axioms which one might want to pose
ocasionally.
Strong homotopy. If two pairs of Γ-equivariant coarse maps are generalized Γ-equivariantly
coarsely homotopic, then they induce the same maps on (co-)homology. That is, the (co-)homo-
logy theory factors through the generalized homotopy category of C.
Flasqueness. The Γ-equivariant coarse (co-)homology theory vanishes on all flasque spaces in C.
Coronality.5 The Γ-equivariant coarse (co-)homology theory vanishes on all relatively compact
spaces in C.
5This axiom is called the large scale axiom in [27, Definition 3.1].
Equivariant Coarse (Co-)Homology Theories 17
Mitchener has shown, that his coarse homology theories are completely determined on coarse
CW-complexes by the values on the ray N and the one-point space, and the same is true of
course for our non-equivariant homology theories, as long as the admissible category C is large
enough to accomodate all the constructions needed. Hence, it is not a good idea to pose both
the flasqueness and the coronality axiom at the same time.
However, all meaningful examples satisfy exactly one of them. Which one we take depends
on how the particular (co-)homology theory should be interpreted: Some coarse (co-)homology
theories arise as coarsifications of locally finite homology or compactly supported cohomology
theories (cf. Section 5.2). Those topological (co-)homology theories often vanish on spaces of
the form X × [0,∞) and as a result the flasqueness axiom will be satisfied (see Section 5.3, in
particular Lemma 5.17).
Other coarse (co-)homology theories are better interpreted as (co-)homology of some (hypo-
thetical) corona. If these hypothetical (co-)homology groups are unreduced ones, then it is the
coronality axiom which is satisfied, because bounded spaces have empty coronas. And if these
hypothetical (co-)homology groups are reduced ones, then it is the flasqueness axiom again.
3.1 Flasqueness implies homotopy invariance
It is well-known that vanishing on flasque spaces can be used to show homotopy invariance of a
functor (see [36, Theorem 9.8], [19, Theorem 11.2], [40, Proposition 3.10] and [6, Proposition 4.16];
similar is [22, Proposition 12.4.12]) and the upcoming Lemma 3.6 is an instance of this principle.
This fact is important, because in some major examples (cf. Sections 4.3 and 4.4) vanishing on
flasque spaces can be shown rather directly by performing certain Eilenberg swindles. Note that
the three notions of coarse homotopy, flasqueness and Eilenberg swindle are closely related in
the sense they all feature the natural numbers (or integers) in an essential way. Therefore, the
same arguments cannot show strong homotopy invariance, where the homotopy parameter runs
through a closed interval.
But first we state an auxiliary lemma. Given any coarse Γ-space X and a Γ-equivariant
bornological function ρ : X → N we define the coarse Γ-spaces
Xρ := {(x, n) ∈ X × Z | n ≥ ρ(x)}, X0 := X × N,
Xρ := {(x, n) ∈ X × Z | n ≤ ρ(x)}, X0 := X ×−N,
Xρ
ρ := {(x, n) ∈ X × Z | n = ρ(x)}, X0
0 := X × {0},
Xρ
0 := {(x, n) ∈ X × Z | 0 ≤ n ≤ ρ(x)}
equipped with the subspace coarse structure of X × Z.
Lemma 3.5. Assume that the admissible category C has the following property: Whenever
(X,A) is an object of C and ρ : X → N \ {0} is a Γ-equivariant bornological function, then C also
contains objects
(
X
ρ
0, A
ρ
0
)
and
(
X
ρ
ρ, A
ρ
ρ
)
satisfying the following properties:
• Aρ
0 ⊂ X
ρ
0 ⊂ X ×N and A
ρ
ρ ⊂ X
ρ
ρ ⊂ X × (N \ {0}) carrying the subspace coarse structure of
the product coarse structure,
•
(
X
ρ
ρ, A
ρ
ρ
)
⊂
(
X
ρ
0, A
ρ
0
)
,
•
(
X
ρ
0, A
ρ
0
)
and
(
X
ρ
ρ, A
ρ
ρ
)
contain
(
Xρ
0 , A
ρ
0
)
and
(
Xρ
ρ , A
ρ
ρ
)
, respectively, as coarsely dense
subspaces.
Then any co- or contravariant functor from C to the category of abelian groups that turns the
inclusions i0 : (X,A) ∼=
(
X0
0 , A
0
0
)
→
(
X
ρ
0, A
ρ
0
)
and iρ :
(
X
ρ
ρ, A
ρ
ρ
)
→
(
X
ρ
0, A
ρ
0
)
as well as the
projections p :
(
X
ρ
0, A
ρ
0
)
→ (X,A) into isomorphisms is homotopy invariant.
18 C. Wulff
All of the admissible categories mentioned earlier have this property, where in most cases we
can take
(
X
ρ
ρ, A
ρ
ρ
)
=
(
Xρ
ρ , A
ρ
ρ
)
and
(
X
ρ
0, A
ρ
0
)
=
(
Xρ
0 , A
ρ
0
)
, except for the categories obtained from
proper metric spaces or topological coarse spaces, in which case we have to take the closures
of Xρ
ρ , A
ρ
ρ, X
ρ
0 and Aρ
0 in X × N. This explains the notation.
Proof. Let (X,A) and (Y,B) be two objects in C and let H : (X,A) × Z → (Y,B) be a Γ-
equivariant coarse homotopy between f and g with associated Γ-equivariant bornological maps
ρ± : X → Z. We may assume ρ− ≡ 0 and ρ := ρ+ : X → N without loss of generality. Otherwise
we simply replace ρ− by min{ρ−,−1} and ρ+ by max{ρ+, 1} and cut the homotopy along X×{0}
into two homotopies of the above type.
Using the assumptions on
(
X
ρ
0, A
ρ
0
)
and
(
X
ρ
ρ, A
ρ
ρ
)
, we can find a coarse map H :
(
X
ρ
0, A
ρ
0
)
→
(Y,B) which agrees with H on
(
Xρ
0 \X
ρ
ρ, A
ρ
0 \A
ρ
ρ
)
, in particular on
(
X0
0 , A
0
0
)
, and with g ◦ p on(
X
ρ
ρ, A
ρ
ρ
)
. The claim now follows by applying the functor to f ◦p◦i0 = H ◦i0 and g◦p◦iρ = H ◦iρ
and exploiting the isomorphism hypotheses. ■
Lemma 3.6. Assume that the admissible category C has the following property: Associated to
each X = (X,∅) in C and each Γ-equivariant bornological function ρ : X → N \ {0} there are
coarse subspaces Xρ, X
ρ
of X × Z with the following properties:
• Xρ and X
ρ
contain Xρ and Xρ, respectively, as coarsely dense subspaces,
• the sets X
ρ
ρ := Xρ ∩X
ρ
and X
ρ
0 := X0 ∩X
ρ
together with the corresponding sets for A ⊂ X
satisfy the assumptions of Lemma 3.5,
• all pairs of coarse Γ-spaces that can be formed from X × Z, X0, X
0, X0
0 , Xρ, X
ρ
, X
ρ
0
and X
ρ
ρ are objects in C.
If EXΓ
∗ or EX∗
Γ satisfies the exactness, excision and flasqueness axioms, then it also satisfies the
homotopy axiom.
Again, the hypothesis is satisfied by all admissible the categories mentioned earlier. The proof
is essentially a Γ-equivariant version of the proofs of the statements cited at the beginning of
this subsection.
Proof. We claim that the functor EXΓ
∗ or EX∗
Γ satisfies the hypotheses of the preceding lemma.
Using the naturality of the long exact sequences under the maps i0, iρ and p and the five-lemma,
we can furthermore restrict ourselves to the absolute case A = ∅.
Now, as EXΓ
∗ or EX∗
Γ satisfies the exactness and excision axioms, it also has Mayer–Vietoris
sequences which can be applied to the decompositions X × Z = X0 ∪X0, X × Z = X0 ∪X
ρ
and
X × Z = Xρ ∪X
ρ
. It furthermore vanishes on the flasque spaces X0, X
0, Xρ, X
ρ
and therefore
the Mayer–Vietoris sequences simplify to isomorphisms
EXΓ
p
(
X0
0
)
= EXΓ
p
(
X0 ∩X0
) ∼= EXΓ
p+1(X × Z),
EXΓ
p
(
X
ρ
0
)
= EXΓ
p
(
X0 ∩X
ρ) ∼= EXΓ
p+1(X × Z),
EXΓ
p
(
X
ρ
ρ
)
= EXΓ
p
(
Xρ ∩X
ρ) ∼= EXΓ
p+1(X × Z)
and analogously for EX∗
Γ. These isomorphisms are natural and therefore the inclusion maps i0:
X0
0 → X
ρ
0 and iρ : X
ρ
ρ → X
ρ
0 induce isomorphisms. Up to the canonical isomorphism X → X0
0 in
the category of coarse spaces, the map p is a one sided inverse to i0. Thus, it induces a one sided
inverse to the isomorphism induced by i0 and must itself be an isomorphism. ■
Equivariant Coarse (Co-)Homology Theories 19
4 Examples
In Section 5 we will describe a procedure to construct a wide class of coarse (co-)homology
theories out of generalized (co-)homology theories for σ-locally compact spaces. Therefore we
shall limit ourselves in this section to a few examples which can be defined by different means.
The first two, ordinary coarse homology (Section 4.1) and cohomology (Section 4.2), are
defined by constructing rather elementary (co-)chain complexes. Note that we have implemented
the Γ-action in a very naive way by considering Γ-invariant chains or dividing out the Γ-action
on the cochain complexes, respectively. This is sufficient for the purpose of giving examples, but
in real life it would be better to develop a more sophisticated Γ-equivariant generalization in the
cohomological case.
For example, one well-known way of defining equivariant cohomology of a topological space X
with action of a topological group Γ is to take a non-equivariant cohomology theory E∗ and
evaluate it on the homotopy quotient:
E∗
Γ(X) := E∗(X ×Γ EΓ).
Question 4.1. Does an analogous construction also work for coarse geometry? And if yes, what
is the coarse geometric analogue of the classifying space EΓ?
An answer to this question would give rise to a wide variety of equivariant cohomology theories.
In Sections 4.3 and 4.4 we show how the K-theories of stable Higson coronas and Roe algebras
fit into our set-up. Here, there are well-established ways of implementing the Γ-actions.
4.1 Ordinary coarse homology
Among all coarse homology and cohomology theories, the ordinary coarse homology is the most
elementary non-trivial one that one could possibly think of. In spirit, it is the obvious dual to
Roe’s coarse cohomology which we will review in the following subsection. In a special case, its
definition has already been given in [46, p. 453], and the general version has been developed in
[6, Section 6.3]. The equivariant version can be found in [7, Section 7]. There is actually no need
at all to pass to a smaller admissible category than the one of coarse Γ-spaces.
In the following, given a coarse space X we always consider Xp+1 as equipped with the (p+1)-
fold product coarse structure. In particular, the penumbras of the multidiagonal ∆X,p+1 :={
(x, . . . , x) ∈ Xp+1 | x ∈ X
}
are those subsets P ⊂ Xp+1 for which there is an entourage E of X
such that for all (x0, . . . , xp) ∈ P and for all 0 ≤ i, j ≤ p we have (xi, xj) ∈ E.
Definition 4.2. Let X be a coarse space and M an abelian group. For p ∈ N we define
CXp(X;M) as the group of all infinite formal sums c =
∑
x∈Xp+1 mxx such that
• the set supp(c) :=
{
x ∈ Xp+1 | mx ̸= 0
}
is a penumbra of the multidiagonal,
• the set supp(c) ∩K is finite for all bounded K ⊂ Xp+1, or equivalently, for all relatively
compact K.6
For p ∈ Z \ N we define CXp(X;M) := 0. These groups together with the boundary maps
∂ : CXp(X;M)→ CXp−1(X;M) which are defined on the single summands by
∂(x0, . . . , xp) :=
p∑
i=0
(−1)i(x0, . . . , xi−1, xi+1, . . . , xp)
constitute a chain complex. If A ⊂ X is a coarse subspace, then CX∗(A;M) is a subcomplex of
CX∗(X;M) and we define the coarse chain complex CX∗(X,A;M) := CX∗(X;M)/CX∗(A;M).
6Recall that relatively compact subsets are finite unions of bouded subsets by Definition 2.10.
20 C. Wulff
Its homology is the ordinary coarse homology HX∗(X,A;M) of the pair of coarse spaces (X,A)
with coefficients in M .
If Γ is a discrete group acting coarsely on (X,A) from the right and acting on M from the
left, then it also acts on the chain complex CX∗(X,A;M) from the left by
γ
( ∑
x∈Xp+1
mxx
)
:=
∑
x∈Xp+1
(γmx)
(
xγ−1
)
=
∑
x∈Xp+1
(γmxγ)x.
The chain
∑
x∈Xp+1 mxx is Γ-invariant, if γmxγ = mx for all γ ∈ Γ and x0, . . . , xp ∈ X.
This property is preserved under the boundary map and hence we obtain the subcomplex
CXΓ
∗ (X,A;M) ⊂ CX∗(X,A;M) of Γ-invariant chains. Its homology is the Γ-equivariant ordinary
coarse homology HXΓ
∗ (X,A;M).
The chain complex CX∗(X;Z) is very similar to the one for uniformly finite homology of [4,
Section 2], just that we have ommitted the uniform bounds on the coefficients and our bounds
on the number of non-zero coefficients in bounded sets are also not uniform in any way. Due to
these uniformity requirements, uniformly finite homology is not functorial under arbitrary coarse
maps but only under the rough maps.
In contrast, it is straightforward to see that ordinary Γ-equivariant coarse homology with
arbitrary coefficients is functorial under Γ-equivariant coarse maps. We will now show that it
satisfies the exactness, strong homotopy, excision and flasqueness axioms and in particular is
a coarse homology theory in the sense of Definition 3.3.
Lemma 4.3. The ordinary Γ-equivariant coarse homology satisfies the exactness axiom.
Proof. The short exact sequence
0→ CX∗(A;M)→ CX∗(X;M)→ CX∗(X,A;M)→ 0
has a Γ-equivariant split CX∗(X;M)→ CX∗(A;M) (not compatible with the boundary map, of
course) which maps all summands not supported within A to 0. This readily implies that the
short sequence of the subcomplexes
0→ CXΓ
∗ (A;M)→ CXΓ
∗ (X;M)→ CXΓ
∗ (X,A;M)→ 0
is also exact. ■
Lemma 4.4. The ordinary Γ-equivariant coarse homology satisfies the strong homotopy axiom.
In particular, it is also functorial under closeness classes of Γ-equivariant coarse maps.
Proof. Let H : (X × [a, b], EU) → (Y, EY ) be a Γ-equivariant generalized coarse homotopy
between f and g which maps A× [a, b] into B. The outline of the proof is that we are going to
construct a “prism operator” (inspired by the classical prism operator in algbraic topology, cf. [18,
proof of Theorem 2.10]) as a chain homotopy between f∗ and g∗ which should be interpreted as
follows: A “simplex” x = (x0, . . . , xp) times the intervall I must be “sudivided” into simplices of
diameter only depending on the diameter of x, but as the diameter of I is allowed to depend
unboundedly on x, the fineness of the subdivision has to be chosen depending on x. Concretely,
this is done as follows.
For each point x ∈ X we choose a = sx,0 < sx,1 < · · · < sx,kx = b such that (s, sx,j) ∈ Ux
and (s, sx,j+1) ∈ Ux for all s ∈ [sx,j , sx,j+1], j = 0, . . . , kx − 1. Due to Uxγ = Ux, this can be
done in such a way that sx,j = sxγ,j for all γ ∈ Γ, x ∈ X and j = 0, . . . , kx − 1 = kxγ − 1. Note
that although we do not require the action of Γ on X to be isocoarse, we cannot weaken the
equivariance condition on the Ux as described in the paragraph following Definition 2.22.
Equivariant Coarse (Co-)Homology Theories 21
The purpose of these choices is that ((x, s), (x, sx,j)) and ((x, s), (x, sx,j+1)) are contained in
the entourage EU whenever s ∈ [sx,j , sx,j+1]. Consequently, if x1, . . . , xn lie in the same E-ball
PenE({x}) around some x ∈ X for some entourage E ∈ EX and if s ∈
⋂p
i=0[sxi,ji , sxi,ji+1], then
the points
(x0, sx0,j0), . . . , (xp, sxp,jp), (x0, sx0,j0+1), . . . , (xp, sxp,jp+1) (4.1)
all lie in the (E ×̄∆[a,b]) ◦ EU -ball Pen(E×̄∆[a,b])◦EU ({(x, s)}) around (x, s). In other words, if
we start with x = (x0, . . . , xp) from a fixed penumbra of the multidiagonal in Xp+1, then for
each q ∈ N any q-tuple of points with entries from (4.1) lies within a fixed penumbra of the
multidiagonal in (X × I)q.
For each x = (x0, . . . , xp) ∈ Xp+1 let kx := kx0 + · · ·+ kxp and we define
jx(l) = (jx,0(l), . . . , jx,p(l)) ∈
p∏
i=0
{0, . . . , kxi}
for l = 0, . . . , kx and ix(l) ∈ {0, . . . , p} for l = 1, . . . , kx recursively as follows: For l = 0 we
choose jx(0) := (0, . . . , 0). If l ≥ 1, then we let ix(l) be the smallest number in {i ∈ {0, . . . , p} |
jx,i(l − 1) + 1 ≤ kxi} such that
sxix(l),jx,ix(l)(l−1)+1 = min{sxi,jx,i(l−1)+1 | i ∈ {0, . . . , p} ∧ jx,i(l − 1) + 1 ≤ kxi}
and we choose jx,ix(l)(l) := jx,ix(l)(l − 1) + 1 and jx,i(l) := jx,i(l − 1) + 1 for i ̸= ix(l). In other
words: jx(l) is obtained from jx(l − 1) by incrementing the ix(l)-th entry. It is clear that this
procedure stops at l = kx with jkx = (kx0 , . . . , kxp).
Using these choices, we now define for each l = 1, . . . , kx the tuple (written as a column vector
for space reasons)
xl :=
(x0, jx,0(l))
...
(xix(l)−1, jx,ix(l)−1(l))
(xix(l), jx,ix(l)(l − 1))
(xix(l), jx,ix(l)(l))
(xix(l)+1, jx,ix(l)+1(l))
...
(xp, jx,p(l))
∈ (X × I)p+2,
that is, the entries are exactly the pairs (xi, jx,i(l)) = (xi, jx,i(l − 1)) for i ̸= ix(l) and together
with the two pairs (xix(l), jx,ix(l)(l − 1)) ̸= (xix(l), jx,ix(l)(l)).
The construction was made in such a way that if x lies in a E-penumbra of the multidiagonal
in Xp+1, then xl lies in a E−1
U ◦ (E ×̄∆[a,b]) ◦ EU -penumbra of the multidiagonal in (X × I)p+2.
Note also, that this construction is clearly Γ-equivariant thanks to the invariant choice of the
sx,j : We have kxγ = kx and (xγ)l = (xl)γ for each l = 1, . . . , kx.
The prism operator for the given Γ-equivariant generalized coarse homotopy is now defined as
PH : CXΓ
p (X,A;M)→ CXΓ
p+1(Y,B;M),∑
x∈Xp+1
mxx 7→
∑
x∈Xp+1
mx
kx∑
l=1
(−1)ix(l)H∗(xl).
Because of the above-mentioned properties, the image of this map indeed consists of Γ-invariant
coarse chains. A calculation similar to the one for the classical prism operator known from basic
algebraic topology (cf. [18, proof of Theorem 2.10]) now shows that PH is a chain homotopy
between f∗ and g∗. ■
22 C. Wulff
Lemma 4.5. The ordinary Γ-equivariant coarse homology satisfies the excision axiom.
Proof. We first show surjectivity of the map HXΓ
p (X\C,A\C;M)→ HXΓ
p (X,A;M). An element
of the target is represented by a Γ-equivariant chain c ∈ CXΓ
p (X;M) with ∂c ∈ CXΓ
p−1(A;M).
As supp(c) is a Γ-invariant penumbra of the multidiagonal, the set
E := {(y, z) ∈ X ×X | ∃(x0, . . . , xp) ∈ supp(c), ∃i, j ∈ {0, . . . , p} : xi = y ∧ xj = z} ∪∆X
is a Γ-invariant entourage which contains the diagonal. In particular, all E-penumbras PenE(B)
of Γ-invariant subspaces B are also Γ-invariant, and we have included the diagonal into E to
ensure that PenE(B) ⊃ B.
Then the chain c can be decomposed as c = c1 + c2 with c2 being the Γ-equivariant chain in
CXΓ
p (PenE(A);M) consisting of all those summands of c which are supported in PenE(A)
p+1,
and therefore c1 ∈ CXΓ
p (X \A;M) ⊂ CXΓ
p (X \ C;M). We have
∂c1 = ∂c− ∂c2 ∈ CXΓ
p−1(X \ C;M) ∩ CXΓ
p−1(PenE(A);M)
= CXΓ
p−1(PenE(A) \ C;M)
and hence c1 represents a class in HXΓ
p (X \ C,PenE(A) \ C;M) which is clearly mapped to the
class of c in HXΓ
p (X,A;M) under the inclusion of pairs of coarse Γ-spaces.
Now the surjectivity of the excision map follows by noting that the inclusion (X \C,A \C) ⊂
(X \ C,PenE(A) \ C) is a Γ-equivariant coarse equivalence and evoking the preceeding lemma.
A coarse inverse up to closeness is obtained as follows: The excisiveness condition gives us an
entourage F such that PenE(A) \ C ⊂ PenF (A \ C). Note that by replacing F with
⋂
γ∈Γ Fγ, if
necessary, we can assume that F is Γ-invariant. This works, because E is Γ-invariant. The coarse
inverse can now be defined as being the identity on X \ PenE(A) and mapping PenE(A) \ C to
A \ C in a way which is F -close to the identity and Γ-equivariant.
In order to show injectivity, we start with c ∈ CXΓ
p (X \ C;M) with ∂c ∈ CXΓ
p−1(A \ C;M)
for which there is d ∈ CXΓ
p+1(X;M) such that ∂d− c ∈ CXΓ
p (A;M), because this is equivalent
to c representing an element of the kernel. Decomposing d into d1 ∈ CXΓ
p+1(X \ C;M) and
d2 ∈ CXΓ
p+1(PenE(A);M) as above, where this time E is obtained from the support of d, we see
that
∂d1 − c = ∂d− c− ∂d2 ∈ CXΓ
p (X \ C;M) ∩ CXΓ
p (PenE(A);M)
= CXΓ
p (PenE(A) \ C;M).
This means that the class of c is mapped to zero under
HXΓ
p (X \ C,A \ C;M)→ HXΓ
p (X \ C,PenE(A) \ C;M),
but we have already seen above that the latter map is an isomorphism. The claim follows. ■
Lemma 4.6. The ordinary Γ-equivariant coarse homology satisfies the flasqueness axiom.
Proof. If (X,A) is a flasque coarse Γ-space with Γ-equivariant ϕ as in Definitions 2.14 and 2.23,
then a chain contraction is given by the M -linear extension CXΓ
∗ (X,A;M)→ CXΓ
∗+1(X,A;M)
of the mapping
(x0, . . . , xp) 7→
∑
n∈N
p∑
i=0
(−1)i
(
ϕn+1(x0), . . . , ϕ
n+1(xi), ϕ
n(xi), . . . , ϕ
n(xi)
)
.
The sums are finite on all relatively compact Kp+1, because the image of ϕn does not intersect K
for n large enough and ϕ is close to the identity. Furthermore, equicontrolledness of the ϕn together
with closeness of ϕ to id ensure that the support of c is a penumbra of the multidiagonal. ■
Equivariant Coarse (Co-)Homology Theories 23
4.2 Ordinary coarse cohomology
Roe defined coarse cohomology in [35] only for metric spaces without group action and with
coefficients in R. We generalize the definition below in the obvious way.
It is a bit surprising and disappointing that, although ordinary coarse cohomology is arguably
the most elementary coarse cohomology theory whose groups can be defined on all coarse Γ-spaces,
we nevertheless have to restrict ourselves to countably generated coarsely connected isocoarse
Γ-spaces in order to be able to prove the excision axiom.
Definition 4.7. LetX be a coarse space andM an abelian group. For p ∈ N we define CXp(X;M)
as the group of all functions φ : Xp+1 →M whose support supp(φ) intersects each penumbra of
the multidiagonal in a relatively compact subset. For p ∈ Z\N we define CXp(X;M) := 0. These
groups together with the Alexander–Spanier coboundary maps δ : CXp(X;M)→ CXp+1(X;M),
that is
δφ(x0, . . . , xp+1) :=
p+1∑
i=0
(−1)iφ(x0, . . . , xi−1, xi+1, . . . , xp+1),
constitute a cochain complex. If A ⊂ X is a coarse subspace, then we define the cochain complex
CX∗(X,A;M) as the kernel of the surjective cochain map CX∗(X;M) → CX∗(A;M) and its
cohomology is the ordinary coarse cohomology HX∗(X,A;M) of the pair (X,A) with coefficients
in M .
If Γ is a discrete group acting isocoarsely on (X,A) from the right and acting onM from the left,
then it also acts on the cochain complex CX∗(X,A;M) from the left via (γφ)(x0, . . . , xp) :=
γ(φ(x0γ, . . . , xpγ)). Hence we have a quotient cochain complex CX∗
Γ(X,A;M) :=
Γ\CX∗(X,A;M), i.e., we divide out the subcomplexes generated by γφ− φ for all γ, φ, and we
call its cohomology the Γ-equivariant coarse cohomology HX∗
Γ(X,A;M).
Ordinary Γ-equivariant coarse cohomology is clearly contravariantly functorial under Γ-
equivariant maps between pairs of coarse Γ-spaces.
Lemma 4.8. The ordinary Γ-equivariant coarse cohomology satisfies the exactness axiom.
Proof. Again, this is implied by the fact that the short exact sequence
0→ CX∗(X,A;M)→ CX∗(X;M)→ CX∗(A;M)→ 0
admits a Γ-equivariant split CX∗(A;M)→ CX∗(X;M), defined by extending cocycles by 0 on
all (p+1)-tuples not supported within Ap+1, and hence the quotient complexes also form a short
exact sequence. ■
Lemma 4.9. The ordinary Γ-equivariant coarse cohomology satisfies the strong homotopy axiom.
In particular, it is also contravariantly functorial under closeness classes of coarse maps.
Proof. The proof is essentially dual to the proof of Lemma 4.4. Using exactly the same kx, ix(l)
and xl constructed from an arbitrary x ∈ Xp+1 we can define the cochain homotopy
PH : CXp+1
Γ (X,A;M)→ CXp
Γ(X,A;M),
PH(φ)(x) :=
kx∑
l=0
(−1)ix(l)φ(H∗(xl)).
It does the job. ■
24 C. Wulff
Lemma 4.10. On the admissible category of pairs of countably generated coarsely connected
isocoarse Γ-spaces the ordinary Γ-equivariant coarse homology satisfies the excision axiom.
Proof. Assume for a moment that there is a Γ-equivariant map r : X → X \ C which restricts
to the identity on X \ C, maps C (and hence also A) into A \ C and restricts to a controlled
map on PenE(X \ C) for every penumbra E of X. We claim that r then induces an inverse to
the excision homomorphism on the cohomology groups.
Let φ ∈ CXp(X \ C,A \ C;M) and P be a penumbra of the multidiagonal in Xp+1. Note
that the latter is equivalent to P being contained in
⋃
x∈X(PenE({x}))p+1 for some entourage E
of X. Hence, if one component xi of x = (x0, . . . , xp) ∈ P is not contained in PenE◦E−1(X \ C),
then none of the components is contained in X \ C. In particular, all components of x are
mapped to A \ C by r, implying rp+1(x) ∈ (A \ C)p+1 and therefore r∗φ(x) = 0. This shows
supp(r∗φ)∩P ⊂ (PenE◦E−1(X \C))p+1. By assumption, the restriction of r to PenE◦E−1(X \C)
is a controlled map. Together with the other assumption on r we see that it is close to the
identity on PenE◦E−1(X \ C) and therefore must also be proper, that is, it is even a coarse map.
Now, controlledness of r implies that rp+1(supp(r∗φ) ∩ P ) is a penumbra of the multidiagonal
in X \ C. In addition, it is also contained in supp(φ) and hence it is relatively compact due to
the support condition on φ. Exploiting that r is also proper we conclude that supp(r∗φ) ∩ P is
relatively compact.
We have just shown that r∗ maps CX∗(X \ C,A \ C;M) to CX∗(X,A;M). The map is
clearly compatible with the group action and the coboundary map. Therefore it induces a map
r∗ : HX∗
Γ(X \ C,A \ C;M)→ HX∗
Γ(X,A;M) on cohomology and we claim that it is an inverse
to the excision map. One of the two compositions is trivially the identity, because r restricts to
the identity on X \ C. The other composition is homotopic to the identity on cochain level via
the “prism operator”
P (φ)(x0, . . . , xp−1) :=
p−1∑
l=0
(−1)lφ(x0, . . . , xl, r(xl), . . . , r(xp−1)),
where the proof that this formula really defines a map CXp
Γ(X,A;M) → CXp−1
Γ (X,A;M) is
completely analogous to the above proof that r∗ maps CXp
Γ(X \ C,A \ C;M) to CXp
Γ(X,A;M).
It remains to prove that the map r exists. If X is a countably generated isocoarse Γ-space,
then we can pick a sequence ∆X = E0 ⊂ E1 ⊂ E2 ⊂ · · · of Γ-invariant entourages such that every
entourage of X is contained in one of the En. Using the alternative excisiveness condition (3.1)
we find for each En an entourage Fn such that A ∩ PenEn(X \C) ⊂ PenFn(A \C). We can then
define r by patching together the identity on X \ C = PenE0(X \ C) with maps
PenEn(X \ C) \ PenEn−1(X \ C)→ A \ C
that are Fn-close to the identity. The assumption of coarse connectedness ensures that this
defines r on the whole space X. All the above-mentioned assumptions on r are readily verified. ■
Finally, the flasqueness axiom is proven by dualizing the proof of Lemma 4.6.
Lemma 4.11. The ordinary Γ-equivariant coarse cohomology satisfies the flasqueness axiom.
4.3 The stable Higson corona
The stable Higson corona was introduced in [10]. A function f : X → D from a coarse space X
to a C∗-algebra D is said to
• vanish at infinity if for each ε > 0 there is a relatively compact subset K such that f is
bounded by ε outside of K,
Equivariant Coarse (Co-)Homology Theories 25
• have vanishing variation if for every entourage E of X the function
VarE f : X → [0,∞), x 7→ sup{∥f(x)− f(y)∥ | (x, y) ∈ E}
vanishes at infinity.
Denote by b(X;D) the C∗-algebra of all bounded functions of vanishing variation from X to
D ⊗ K, where K denotes the C∗-algebra of compact operators on a separable infinite dimensional
Hilbert space. Furthermore, we define b
red
(X;D) to be the C∗-algebra of all bounded functions
of vanishing variation from X into the stable multiplier algebra Ms(D) := M(D ⊗ K) of D
such that f(x)− f(y) ∈ D ⊗ K for all x, y ∈ X which lie in the same coarse component. Both
of them contain the ideal B0(X;D ⊗ K) of all bounded functions X → D ⊗ K vanishing at
infinity. The C∗-algebras b(X;D), b
red
(X;D), B0(X;D ⊗ K) have been introduced in [10, proof
of Proposition 3.7] with the additional condition that the functions should also be Borel, but all
statements that are relevant to us still hold without this condition.
The stable Higson corona of X with coefficients in D is the quotient C∗-algebra
c(X;D) := b(X;D)/B0(X;D ⊗ K)
and the reduced stable Higson corona of X with coefficients in D is the quotient C∗-algebra
cred(X;D) := b
red
(X;D)/B0(X;D ⊗ K).
All of the above function algebras are clearly contravariantly functorial under closeness classes of
coarse maps.
If X is moreover a topological coarse space, then there are the sub-C∗-algebras of continuous
functions
C0(X;D ⊗ K) ⊂ B0(X;D ⊗ K), c(X;D) ⊂ b(X;D), cred(X;D) ⊂ b
red
(X;D),
which are contravariantly functorial under continuous coarse maps and the inclusions induce
isomorphisms
c(X;D) ∼= c(X;D)/C0(X;D ⊗ K),
cred(X;D) ∼= cred(X;D)/C0(X;D ⊗ K), (4.2)
see [10, proof of Proposition 3.7]. The right hand sides are the familiar original definitions of the
stable Higson coronas [10, Definitions 3.2 and 5.4].
For our purposes, we need the relative versions:
Definition 4.12. If (X,A) is a pair of coarse spaces, then we define its (unreduced and reduced)
relative stable Higson coronas with coefficients in D as the kernels of the restrictions to A,
c(X,A;D) := ker(c(X;D)→ c(A;D)),
cred(X,A;D) := ker
(
cred(X;D)→ cred(A;D)
)
.
If (X,A) is even a pair of topological coarse spaces, then we also define its (unreduced and reduced)
relative stable Higson compactifications with coefficients in D as the kernels of the restrictions
to A,
c(X,A;D) := ker(c(X;D)→ c(A;D)),
cred(X,A;D) := ker
(
cred(X;D)→ cred(A;D)
)
.
26 C. Wulff
Note that c(X,A;D) = cred(X,A;D) and c(X,A;D) = cred(X,A;D) whenever A meets
all coarse components of X, whereas for A = ∅ we recover c(X;D), cred(X;D), c(X;D)
and cred(X;D). Also, c(X;D) = cred(X,K;D) for any non-empty relatively compact K ⊂ X.
The relative stable Higson coronas are clearly contravariantly functorial under coarse maps of
pairs and the relative stable Higson compactifications are so under continuous coarse maps of
pairs.
Remark 4.13. For coarsely disconnected spaces X, it is debatable what the correct definition of
the reduced stable Higson corona is. One could also ask for f(x)− f(y) ∈ D⊗K for all x, y ∈ X,
not just for those in the same connected component, just like the reduced singular (co-)chain
complexes are obtained from the unreduced ones by augmentation with one copy of Z and not
with one copy for each connected component of the space. As a result, the K-theory of this
reduced version of the stable Higson corona would not satisfy flasqueness (cf. Lemma 4.16), which
is in analogy to reduced singular (co-)homology not vanishing on disjoint unions of more than
one contractible spaces.
Our choice of definition is due to the fact that flasqueness is an indispensable technical tool and
we need a version of this coarse cohomology theory satisfying this axiom. Later on, in Section 5.4,
we will also discuss a more flexible notion of topological reduced coarse (co-)homology, which
also allows for componentwise augmentation.
Note that we would immediately obtain long exact sequences for the K-theory groups
K1−∗(c(−,−;D)) and K1−∗
(
cred(−,−;D)
)
if only the restriction maps to A were surjective.
Unfortunately, this is a rather delicate problem which has only been solved in special cases.
Willett had shown it for pairs of coarsely connected proper metric spaces for the reduced stable
Higson coronas with trivial coefficients D = C [40, Lemma 3.4], but the proof also works for
arbitrary coefficients and the claim for the unreduced stable Higson corona is an easy consequence.
We will show in Lemma 4.14 below how to deduce the generalization to coarsely disconnected
proper metric spaces. Bunke and Engel have discussed a generalization to a bigger class of coarse
spaces, but also could not overcome all dificulties [5, Section 5].
Lemma 4.14. If X is a proper metric space and A ⊂ X, then the restriction maps c(X;D)→
c(A;D) and cred(X;D)→ cred(A;D) are surjective.
Proof. The reduced and unreduced cases are clearly equivalent to each other, and they follow
from surjectivity of the restriction map c(X;D)→ c(A;D), so let us show the latter.
First of all we may assume that A meets all coarse components of X, because the extensions
can simply be chosen to be zero on all coarse components which are disjoint from A.
Second, it suffices to consider the case, where X and A have at most countably many coarse
components. In order to see this, let f ∈ c(A;D). For each R ∈ N and n ∈ N \ {0} we
find a relatively compact subset KR,n ⊂ A such that VarER
f(x) < 1
n for all x ∈ A \ KR,n.
As each KR,n meets only finitely many coarse components of A, their union
⋃
R,nKR,n are
contained in a countable union of coarse components of A, which we denote by A′ ⊂ A, and
let X ′ ⊂ X be the union of the corresponding coarse components of X. For all x ∈ A \ A′ we
therefore have VarER
f(x) < 1
n for all R ∈ N and n ∈ N \ {0}, so that f is a constant function
on the coarse component containing x. Thus, if we know that c(X ′;D)→ c(A′;D) is surjective,
then we can pick any preimage F ′ ∈ c(X ′;D) of f |A′ and extend it to a function F ∈ c(X;D)
which is constant on each coarse component of X \X ′ and such that F |A = f .
Thirdly, we can deduce the statement from the one for coarsely connected spaces as follows.
Let S ⊂
{
n2 | n ∈ N
}
be a set of square numbers which has the same cardinality as the set Π0(A)
of coarse components of A (and hence also X) and let S → A, s 7→ xs be a map which induces a
bijection S → Π0(A). If d : X ×X → [0,∞] is the original coarsely disconnected proper metric,
then we can define the coarsely connected proper metric d′ : X ×X → [0,∞) on X as the largest
Equivariant Coarse (Co-)Homology Theories 27
metric such that d′ ≤ d and d(xs, xt) ≤ |s− t| for all s, t ∈ S. It is then clear that changing the
metric from d to d′ does not change the C∗-algebras c(X;D) and c(A;D).
Finally, for coarsely connected proper metric spaces the claim is proven exactly as [40,
Lemma 3.4]. ■
Lemma 4.15. If X is a proper metric space and C ⊂ A ⊂ X satisfy the coarse excisiveness
condition, then restriction to X \ C yields isomorphism c(X,A;D) ∼= c(X \ C,A \ C;D) and
cred(X,A;D) ∼= cred(X \ C,A \ C;D).
Proof. Injectivity is straightforward: If a function on X represents an element of the kernel,
then its restrictions to both A and X \ C vanish at infinity. These two sets cover X and the
relatively compact subsets of X are exactly the union of relatively compact subsets of A with
relatively compact subsets of X \ C. This implies that the function itself vanishes at infinity,
and therefore represents the zero element.
For surjectivity we use that the right square of the diagram
0 // cred(X,A;D) //
��
cred(X;D) //
��
cred(A;D) //
��
0
0 // cred(X \ C,A \ C;D) // cred(X \ C;D) // cred(A \ C;D) // 0
is a pullback diagram. For coarsely connected spaces and trivial coefficients, this is [40, Lem-
ma 3.3], but the proof goes through for coarsely disconnected spaces and arbitrary coefficients as
well. Furthermore, the same is true for the analogous unreduced version of the diagram. The
claim now follows from an easy diagram chase. ■
Lemma 4.16. For flasque proper metric spaces X, the group K1−∗
(
cred(X;D)
)
vanishes.
Proof. This is the straightforward adaption of [40, Proposition 3.7] (see also [10, Theorem 5.2])
to the case of not necessarily coarsely connected spaces and with coefficients. Let us briefly
recall Willett’s sketch of proof, to see why our definition of the reduced stable Higson corona for
coarsely disconnected spaces is the one fulfilling this property.
Denote by H the infinite dimensional separable Hilbert space on which K := K(H) is modelled,
such that we also haveMs(D) = B(D⊗H). Choose an isometric isomorphism H ∼= ⊕NH and let
Vn : H → H be the isometric inclusion as the n-th summand with respect to this decomposition.
Furthermore, let ϕ : X → X be the coarse map from the definition of flasqueness. For every
f ∈ cred(X;D) one now defines
νf : X →Ms(D), x 7→
∑
n∈N
(idD ⊗Vn) ◦ f(ϕn(x)) ◦ (idD ⊗Vn)∗.
For each two x, y in the same coarse component ofX, the sequences (ϕn(x))n∈N and (ϕn(y))n∈N
stay close to each other and go to infinity. Hence the vanishing variation of f implies that the
differences f(ϕn(x))− f(ϕn(y)) converge in norm to zero for n→∞. As they are also compact,
the difference
νf(x)− νf(y) =
∑
n∈N
(idD ⊗Vn) ◦
(
f(ϕn(x))− f(ϕn(y))
)
◦ (idD ⊗Vn)∗
is compact, too. Note that the part of this argument using vanishing variation does not work
for x, y in different coarse components, and that is the reason why this lemma only holds for our
definition of the reduced stable Higson corona and not the alternative mentioned in Remark 4.13.
28 C. Wulff
Furthermore, it is straightforward to check that νf has vanishing variation and that the result-
ing ∗-homomorphism ν : cred(X;D)→ cred(X;D) descends to a ∗-homomorphism ν : cred(X;D)
→ cred(X;D) on the corona. The proof is then completed by the following Eilenberg swindle:
one shows that on K-theory we have ν∗(x) = x+ ν∗(x) for all x ∈ K1−∗
(
cred(X;D)
)
, so the latter
group vanishes. ■
Theorem 4.17. The contravariant functors K1−∗
(
cred(−,−;D)
)
and K1−∗(c(−,−;D)) are co-
arse cohomology theories on the category of proper metric spaces and coarse maps. The first one
satisfies the flasqueness axiom and the second one the coronality axiom.
Proof. The long exact sequences and excision are immediate from Lemmas 4.14 and 4.15. The
coronality axiom for the second one is clear, because c(K;D) = 0 for all relatively compact
spaces K.
The flasqueness axiom for the first one follows from 4.16 for the absolute groups together
with the long exact sequence for the relative groups. Flasqueness also implies the homotopy
invariance of the first functor by Lemma 3.6.
The homotopy invariance of the second functor is a bit tricky, but it can also be deduced
from the flasqueness of the first one as follows. The proof of Lemma 3.6 worked by showing
that the hypothesis of Lemma 3.5 is satisfied. Thus, we already know that the first functor
K1−∗
(
cred(−,−;D)
)
satisfies this hypothesis and it remains to show that the second functor
K1−∗(c(−,−;D)) does so as well. Given any pair (X,A) of proper metric spaces, we denote by
Π0 := Π0(X,A) the set of those coarse components of X which do not contain any points of A.
Then there are canonical short exact sequences
0→ c(Y,B;D)→ cred(Y,B;D)→
∏
Π0
Ms(D)
D ⊗ K
→ 0
for all those pairs (Y,B) of proper metric spaces derived from (X,A) which appear in Lemma 3.5.
Consider the maps between the associated long exact K-theory sequences induced by the
maps i0, iρ and p. For i0 this yields a diagram
K2−p
(
cred
(
X
ρ
0;D
))
//
i∗0 ∼=
��
K2−p
(∏
Π0
Ms(D)
D⊗K
)
// K1−p
(
c
(
X
ρ
0;D
))
//
i∗0
��
K2−p
(
cred(X;D)
)
// K2−p
(∏
Π0
Ms(D)
D⊗K
)
// K1−p(c(X;D)) //
// K1−p
(
cred
(
X
ρ
0;D
))
//
i∗0 ∼=
��
K1−p
(∏
Π0
Ms(D)
D⊗K
)
// K1−p
(
cred(X;D)
)
// K1−p
(∏
Π0
Ms(D)
D⊗K
)
and the five-lemma implies that the middle verticle map is an isomorphism. Similarly, the
maps iρ and p induce isomorphisms under the second functor. ■
We now briefly discuss two equivariant generalizations for pairs of coarsely connected proper
metric Γ-spaces, where we assume that the coefficient C∗-algebra D is a Γ-C∗-algebra. In this
case, there are canonical actions of Γ on cred(X,A;D) and c(X,A;D) in which γ[f ] is represented
by the function γf : x 7→ γ(f(xγ)). Recall that we only consider discrete groups (which even
have to be countable, because they act properly on coarsely connected proper metric spaces) and
thus do not have to worry about continuity of the group actions.
Equivariant Coarse (Co-)Homology Theories 29
Theorem 4.18. If ⋊Γ denotes an exact crossed product functor in the sense of [2, Definition 3.1],
then the functors K1−∗
(
cred(−,−;D) ⋊ Γ
)
and K1−∗(c(−,−;D) ⋊ Γ) are Γ-equivariant coarse
cohomology theories satisfying the flasqueness or the coronality axiom, respectively.
Proof. Exactness and excision again follow directly from Lemmas 4.14 and 4.15, and coronality
of the unreduced versions is again clear.
The flasqueness axiom for the reduced version is proven exactly as in the non-equivariant case,
because the ∗-homomorphism ν in the proof of Lemma 4.16 will now be equivariant.
Homotopy invariance of the reduced version follows. For the unreduced version we prove it
just as in the proof of the previous theorem, where the induced Γ-action on Π0 := Π0(X,A) and
the action on D turn
∏
Π0
Ms(D)
D⊗K into a Γ-C∗-algebra. ■
The above equivariant version is not the most sophisticated. According to the work of Emerson
and Meyer [11, 12], better choices are Ktop
1−∗(Γ, c(−,−;D)) and Ktop
1−∗
(
Γ, cred(−,−;D)
)
. We do
not need to recall their definiton, because the only thing relevant to us is that there are natural
isomorphisms
Ktop
∗ (Γ, C) ∼= K∗((C ⊗max P)⋊max Γ) ∼= K∗((C ⊗max P)⋊red Γ), (4.3)
which hold for both the reduced and full crossed product functor and for any Γ-C∗-algebra C
(see [11, Theorem 22], which is based on [26, Theorems 5.2 and 10.2]7). Here, P denotes
a certain Γ-C∗-algebra which supports the so-called Dirac morphism D ∈ KKΓ(P,C). Exactly
the same proof as for the previous theorem, just with (−⊗max P)⋊max instead of ⋊, shows the
following.
Theorem 4.19. The functors Ktop
1−∗
(
Γ, cred(−,−;D)
)
and Ktop
1−∗(Γ, c(−,−;D)) are Γ-equivariant
coarse cohomology theories satisfying the flasqueness or the coronality axiom, respectively.
Let us briefly discuss the relation between all of these theories. First of all, the Baum–Connes
assembly map yields natural transformations
Ktop
1−∗
(
Γ, cred(−,−;D)
)
→ K1−∗
(
cred(−,−;D)⋊max Γ
)
→ K1−∗
(
cred(−,−;D)⋊ Γ
)
,
Ktop
1−∗(Γ, c(−,−;D))→ K1−∗(c(−,−;D)⋊max Γ)→ K1−∗(c(−,−;D)⋊ Γ),
which are compatible with the connecting homomorphisms.
Second, there is a commutative diagram
0 //
∏
Π0
(D ⊗ K) //
��
∏
Π0
Ms(D) //
��
∏
Π0
Ms(D)
D⊗K
// 0
0 // c(X,A;D) // cred(X,A;D) //
∏
Π0
Ms(D)
D⊗K
// 0
with exact rows, where Π0 := Π0(X,A) again denotes the set of those coarse components of X
which do not contain any points of A. The K-theory of
∏
Π0
Ms(D),
(∏
Π0
Ms(D)
)
⋊ Γ and(∏
Π0
(Ms(D))⊗max P
)
⋊max Γ, vanishes by an Eilenberg swindle and hence we can easily read
7Separability of the C∗-algebras is a standing assumption in the latter reference, but the isomorphisms here
also hold for non-separable C, because we can take the direct limit over all separable sub-C∗-algebras and exploit
that all constructions in (4.3) are co-continuous under such.
30 C. Wulff
off the following long exact sequences in K-theory:
...
��
...
��
...
��
K1−∗
(∏
Π0
(D ⊗ K)
)
��
K1−∗
((∏
Π0
(D ⊗ K)
)
⋊ Γ
)
��
Ktop
1−∗
(
Γ,
∏
Π0
(D ⊗ K)
)
��
K1−∗(c(X,A;D))
��
K1−∗(c(X,A;D)⋊ Γ)
��
Ktop
1−∗(Γ, c(X,A;D))
��
K1−∗
(
cred(X,A;D)
)
��
K1−∗
(
cred(X,A;D)⋊ Γ
)
��
Ktop
1−∗
(
Γ, cred(X,A;D)
)
��
K−∗
(∏
Π0
(D ⊗ K)
)
��
K−∗
((∏
Π0
(D ⊗ K)
)
⋊ Γ
)
��
Ktop
−∗
(
Γ;
∏
Π0
(D ⊗ K)
)
��...
...
...
(4.4)
The first of these three sequences decomposes for an unbounded and coarsely connected X and
empty A into the short exact sequence
0→ K1−∗(D)→ K1−∗(c(X;D))→ K1−∗
(
cred(X;D)
)
→ 0
(cf. [10, Lemma 3.10]), but the proof does not generalize to the equivariant case, because it
involves evaluation ∗-homomorphisms at points of X, which are in general not equivariant.
The closest we come to an equivariant generalization is the following: If X contains a flasque
subspace B which is contained in the union of the components in Π0(X,A) and contains points
from all components in Π0(X,A) (in other words, the inclusion B ⊂ X induces a bijection
between the set Π0(B) of coarse components of B and Π0(X,A)), then the above long exact
sequences are mapped by restriction to the corresponding ones for B. But then the arrows
leaving the reduced terms in the sequences for (X,A) factor through the corresponding terms
for B, which vanish because of flasqueness. Thus, we get a decomposition of the three long exact
sequences into short exact sequences in this case, too.
Third, as the long exact sequences (4.4) indicate, there is also a way of obtaining the reduced
groups from the unreduced stable Higson corona with the help of a mapping cone construction.
In fact, it is this construction which completely establishes the analogy to reduced cohomology
theories as we will see later in Sections 5.4 and 6.3 (keeping Remark 4.13 in mind). Recall that
the mapping cone of a ∗-homomorphism f : A→ B is defined as the C∗-algebra
Cone(f) := {(a, β) ∈ A× C0((0, 1], B) | β(1) = f(a)}
and if f is surjective, then the inclusion ker(f)→ Cone(f), a 7→ (a, 0) induces an isomorphism
on K-theory.
Lemma 4.20. Let Π0 := Π0(X,A), again, denote the set of coarse components of X which are
disjoint from A. Then there are canonical (up to a sign) natural isomorphisms
K1−∗
(
cred(X;D)
) ∼= K−∗
(
Cone
(∏
Π0
(D ⊗ K)→ c(X;D)
))
,
K1−∗
(
cred(X;D)⋊ Γ
) ∼= K−∗
(
Cone
(∏
Π0
(D ⊗ K)→ c(X;D)
)
⋊ Γ
)
,
Ktop
1−∗
(
Γ, cred(X;D)
) ∼= Ktop
−∗
(
Γ,Cone
(∏
Π0
(D ⊗ K)→ c(X;D)
))
under which the long exact sequences (4.4) are isomorphic to the long exact sequences induced by
0→ C0(0, 1)⊗ c(X;D)→ Cone(
∏
Π0
(D ⊗ K)→ c(X;D))→
∏
Π0
(D ⊗ K)→ 0.
Equivariant Coarse (Co-)Homology Theories 31
Proof. In order to reuse the exact same proof for different algebras later on (Lemma 6.4), we
write I := c(X;D), A := cred(X;D), B :=
∏
Π0
Ms(D) and J :=
∏
Π0
(D ⊗ K). The decisive
properties are that I is an ideal in A, J is an ideal in B, K∗(B) = K∗(B ⋊ Γ) = Ktop
∗ (Γ, B) = 0
(because of an Eilenberg swindle argument) and there is a ∗-homomorphism ι : B → A which
induces an isomorphism B/J ∼= A/I. In particular, ι restricts to ι′ : J → I.
We furthermore let π : B → B/J be the quotient homomorphism and consider the diagram
with exact rows
0 // C0(0, 1)⊗ I // C0(0, 1)⊗A //
��
C0(0, 1)⊗B/J //
��
0
0 // C0(0, 1)⊗ I // Cone(ι) // Cone(π) // 0
0 // C0(0, 1)⊗ I // Cone(ι′)
OO
// J
OO
// 0.
Both vertical arrows on the right are known to induce isomorphisms on K-theory. They are
the ones implementing the connecting homomorphism K1−∗(B/J)
∼=−→ K−∗(J). Therefore, the
five-lemma shows that the diagram induces isomorphisms between the long exact sequences in
K-theory associated to the rows. This is exactly the first isomorphism of the claim.
For the other two, we apply the functors −⋊ Γ and (−⊗max P)⋊max Γ to the whole diagram
and note that all the properties mentioned above still hold, because the crossed products are
exact and compatible with tensor products, hence also continuous and compatible with the cone
construction. ■
And last, there is another relation between the reduced and unreduced groups which is based
upon the idea that an unreduced cohomology can be obtained by taking reduced cohomology of
the space augmented by a distinct basepoint (cf. [43, Section 6]). But with our Definition 4.12
and in the light of Remark 4.13 we have to add not one distinct point but instead the whole
set of coarse components Π0(X) as isolated points, in some sense. To this end, assume that
there exists a relatively compact subspace K ⊂ X which contains points from all connected
components of X. Note that this might not be the case if Γ is infinite. We define the space
X→ := X∪KK×N equipped with the obvious metric. The subspace K×N is coarsely equivalent
to Π0(X)× N, considered as the coarsely disjoint union of copies of N, and each of these copies
should be interpreted as a newly added point at infinity. We note that (X,K) ⊂ (X→,K ×N) is
an excision, K × N is flasque and c(X,K;D) = c(X;D). Therefore, the long exact sequence for
the pair (X→,K × N) gives rise to isomorphisms
K1−∗(c(X;D)) = K1−∗(c(X,K;D)) ∼= K1−∗(c(X
→,K × N;D)) ∼= K1−∗
(
cred(X→;D)
)
and analogously for the two equivariant versions.
4.4 K-theory of the Roe algebra
The Roe algebra is a C∗-algebra C∗(X) that can be constructed for every coarse space X (cf. [37,
Section 4.4]), but it is usually only dealt with in the case of proper metric spaces. As most of
the references only consider the latter special case, we shall stick to it.
Let us briefly recall the broad outlines of its construction as presented, e.g., in [22, Section 6.3].
Starting with a representation ρ of C0(X) on a separable Hilbert space H which is non-degenerate
(meaning that ρ(C0(X))H is dense in H), the Roe algebra C∗(X, ρ) is defined as the norm
closure of the ∗-subalgebra of B(H) consisting of all operators T satisfying the following two
properties:
• T is called locally compact, if ρ(f)T, Tρ(f) are compact operators on H for all f ∈ C0(X).
32 C. Wulff
• T has finite propagation, if there is R ≥ 0 such that ρ(f)Tρ(g) = 0 whenever the distance
between the supports of f, g ∈ C0(X) is at least R. The infimum of all R ≥ 0 with this
property is called the propagation prop(T ) of T .
This C∗-algebra depends on the choice of H and ρ, but if the representation is big enough in
a certain sense (it is “ample”, i.e., it is non-degenerate and ρ−1(K(H)) = {0}), then its K-theory
K∗(C
∗(X, ρ)) is independent from the choices up to canonical isomorphism. Therefore, the
representation ρ is usually omitted from the notation with the understanding that one suitable
representation has been chosen for each X.
Now, there are two well-known generalizations of the Roe algebra. The first is the equivariant
Roe algebra in the case that a countable discrete group Γ acts properly and isometrically on the
coarsely proper metric space X. Here one chooses in addition to the non-degenerate representa-
tion ρ also a representation u of the group Γ on H by unitaries such that u(γ)ρ(f)u(γ)∗ = ρ(γ ·f)
for all γ ∈ Γ and f ∈ C0(X). The triple (H, ρ, u) is called a Γ-X-module. Then the equivariant
Roe algebra C∗
Γ(X, ρ, u) is the norm closure of the C∗-algebra of all locally compact operators of
finite propagation in B(H) that are equivariant with respect to u. Again, the K-theory groups
K∗(C
∗
Γ(X, ρ, u)) are independent up to canonical isomorphism from the Γ-X-module (H, ρ, u) if
the latter has been chosen sufficiently large and then abbreviated by K∗(C
∗
Γ(X)). A detailed
exposition about equivariant Roe algebras can be found in [41].
The second generalization is to introduce a (possibly graded) coefficient C∗-algebra D into the
Roe algebra, cf. [17, 19, 44]. This is done by replacing the Hilbert space H by a right Hilbert
module H over D and the non-degenerate representation ρ is by adjointable operators on H. The
K-theory of the resulting C∗-algebra C∗(X, ρ,H;D) is again unique up to canonical isomorphism
if ρ and H have been chosen sufficiently large and in this case ρ will be ommited from the notation.
Here, sufficient largeness means that H ∼= H ⊗D and ρ = ρ′⊗ idD for an ample representation ρ′
on a separable Hilbert space H [19, Definition 4.5].
It is possible to unify the two generalizations to equivariant Roe algebras C∗
Γ(X, ρ, u,H;D)
with coefficients in a (graded) Γ-C∗-algebra D. In this case, u also has to be a representation
by adjointable operators on H which is compatible with the Γ-action on D in the sense that
u(γ)(ξd) = (u(γ)ξ)(γd) for all ξ ∈ H, d ∈ D and γ ∈ Γ. Again, we obtain unambiguous K-theory
groups K∗(C
∗
Γ(X;D)) if we start with sufficiently large H, ρ, u as in the equivariant case and then
simply use their tensor product with D, i.e., H := H ⊗D with obvious induced representations
of C0(X) and Γ.
The proofs of the following statements can be found in the above-mentioned sources in
the special cases just described, but the proofs generalize to the general case of equivariant
Roe-algebras with coefficients.
If A ⊂ X is a Γ-invariant closed subspace of the proper metric Γ-space X, then C∗
Γ(X;D)
contains an ideal C∗
Γ(A ⊂ X;D) whose K-theory is canonically isomorphic to the K-theory of
C∗
Γ(A;D). It is the norm closure of all T ∈ C∗
Γ(X;D) which are supported in an R-neighborhood
of A for some R ≥ 0, that is, all those T for which ρ(f)T = Tρ(f) = 0 for all f ∈ C0(X)
with dist(supp(f), A) ≥ R. The quotient C∗-algebra C∗
Γ(X,A;D) := C∗
Γ(X;D)/C∗
Γ(A ⊂ X;D)
is called the Γ-equivariant relative Roe algebra. We have C∗
Γ(∅ ⊂ X;D) = 0 and hence
C∗
Γ(X,∅;D) = C∗
Γ(X;D). The long exact sequences
· · · → Kp(C
∗
Γ(A;D))→ Kp(C
∗
Γ(X;D))→ Kp(C
∗
Γ(X,A;D))→ Kp−1(C
∗
Γ(A;D))→ · · ·
are immediate.
The K-theory of the Γ-equivariant relative Roe algebra with coefficients in D is functorial
under Γ-equivariant coarse maps between pairs of proper metric Γ-spaces and the non-boundary
maps in the long exact sequence above are special cases of this functoriality under the inclusion
maps (A,∅)→ (X,∅)→ (X,A).
Equivariant Coarse (Co-)Homology Theories 33
For excision, we note that for an excision (A \ C,X \ C)→ (X,A) the left square in
0 // C∗
Γ(A \ C;D) //
��
C∗
Γ(X \ C;D) //
��
C∗
Γ(X \ C,A \ C;D) //
��
0
0 // C∗
Γ(A;D) // C∗
Γ(X;D) // C∗
Γ(X,A;D) // 0
is simultaneously a pull-back and a push-out diagram (cf. [22, p. 156] for the non-equivariant
case without coefficients, but the general case is proven in exactly the same way) and therefore
the excision isomorphism already holds at the level of Γ-C∗-algebras.
Last but not least, the K-theory of the Roe algebra vanishes on flasque spaces by an Eilenberg
swindle (cf. [36, Proposition 9.4], which is also readily generalized to the equivariant case with
coefficients) and therefore is also satisfies the homotopy axiom. Even more, the methods developed
by Higson and Roe in [20] can be adapted to show the following.
Lemma 4.21. The K-theory of the Roe algebra satisfies the strong homotopy axiom.
Proof. Let H : (X × [0, 1], EU )→ (Y, dY ) be a Γ-equivariant generalized coarse homotopy which
maps A × [0, 1] into B. In order to stay within the world of proper metric Γ-spaces, we may
use the metric defined as follows instead of the coarse structure EU . Without loss of generality
we can assume that the metric dX on X is discrete and make a Γ-invariant choice of cx ≥ 1 for
all x ∈ X such that |s − t| ≤ 1
cx
=⇒ (s, t) ∈ Ux. The largest metric d on X × [0, 1] with the
property
∀x, y ∈ X, s, t ∈ [0, 1] : d((x, s), (y, t)) ≤ dX(x, y) + min{cx, cy} · |s− t|
is proper and Γ-invariant and H is still coarse as a map (X × [0, 1], d)→ (Y, dY ).
Let et : (X,A)→ (X,A)× [0, 1], x 7→ (x, t) denote the canonical inclusions. It is then sufficient
to prove that e0 and e1 induce the same map
(e0)∗ = (e1)∗ : K∗(C
∗
Γ(X,A;D))→ K∗(C
∗
Γ((X,A)× [0, 1];D)),
where the right hand side is defined using the above metric d on X× [0, 1]. In the non-equivariant,
non-relative case without coefficients (Γ = 1, A = ∅, D = C) this has been proved in [20,
Sections 5 and 6], albeit with a slightly different metric on X × [0, 1]. The proof relies heavily
on calculations with idempotents, which is not adequate for the relative case A ̸= ∅ and for
non-unital coefficient C∗-algebras D. Therefore, we use the same tools but apply them a bit
differently to obtain the general result.
Recall the following technique to define elements of the K-theory of a C∗-algebra C from [20,
Section 4]: LetM(C) denote its multiplier algebra and define D(C) := {(a, b) ∈ M(C) | a− b
∈ C}. Then there is a split short exact sequence
0 // K∗(C)
j∗ // K∗(D(C)) π∗
// K∗(M(C)) //
s∗qq
0,
where j : c 7→ (c, 0), π : (a, b) 7→ b and s : b 7→ (b, b). An element of K∗(C) can then be defined as
the image of an element of K∗(D(C)) under Θ := (j∗)
−1 ◦ (id−s∗ ◦ π∗).
Furthermore, we recall that the asymptotic algebra of a C∗-algebra C is defined as A(C) :=
Cb([1,∞);C)/C0([1,∞);C) and an asymptotic morphism from another C∗-algebra B to C is
a ∗-homomorphism ψ : B → A(C). As the C∗-algebra C0([1,∞);C) is contractible, such an
asymptotic morphism induces a homomorphism
ψ∗ : K∗(B)
Ψ∗−−→ K∗(A(C)) ∼= K∗(Cb([1,∞);C))
(ev1)∗−−−−→ K∗(C).
34 C. Wulff
The plan is to construct an algebra homomorphism
Ψ: C
(
S1
)
⊗ C∗
Γ(X,A;D)→ A
(
C[0, 1]⊗ C
(
S1
)
⊗D
(
C∗
Γ((X,A)× [0, 1];D))
))
. (4.5)
It will not be a ∗-homomorphism (and therefore strictly speaking not an asymptotic morphism),
but as algebraic and topological K-theory agree in degree 0 we will still obtain an induced map
on K0.
8 Since K0(C(S
1)⊗ C) ∼= K0(C)⊕K1(C) in topological K-theory for any C∗-algebra C,
this induced map decomposes into two maps
Ψ∗ : Ki(C
∗
Γ(X,A;D))→ Ki(C[0, 1]⊗D(C∗
Γ((X,A)× [0, 1];D))), i = 0, 1.
Now, the evaluation ∗-homomorphisms evt : C[0, 1]→ C for t ∈ [0, 1] are all homotopic to each
other and therefore all of the maps
Θ ◦ (evt)∗ ◦Ψ∗ : Ki(C
∗
Γ(X,A;D))→ Ki(C
∗
Γ((X,A)× [0, 1];D))
are equal. For the specific Ψ which we are about to define and t = 0, 1, these maps are exactly
(e0)∗ and (e1)∗, respectively. The claim follows.
In order to carry this out, assume that C∗
Γ(X;D) has been constructed on the Hilbert module
H := H⊗D for a sufficiently large Γ-X-module (H, ρ, u). Furthermore, let ρ′ : C[0, 1]→ B
(
L2(R)
)
be the representation, where ϕ ∈ C[0, 1] acts on L2(R) by multiplication with
ϕ̃ : s 7→
ϕ(0), s ≤ 0,
ϕ(t), 0 ≤ t ≤ 1,
ϕ(1), t ≥ 1.
Then we choose two copies of the Γ-(X × [0, 1])-module
(
H ⊗ L2(R), ρ⊗ ρ′, u⊗ 1
)
to construct
C∗
Γ(X × [0, 1];D), that is, this C∗-algebra consists of operators on
(
H ⊗ L2(R)⊗D
)2
.
The relevant outcome of [20, Sections 6] can be summarized as follows. There exists a bounded
and equicontinuous family of maps
{
Fp : [0, 1]→ B
(
L2(R)
)}
p∈(0,1] such that
• the operators Fp(s) are Fredholm of index one and Fp(s)
∗ is an inverse to Fp(s) modulo
compact operators,
• for fixed s and varying p, the operators Fp(s) are compact perturbations of one another,
• the propagation of Fp(s) and Fp(s)
∗ with respect to the representation ρ′ is bounded by p,
• Fp(0) is independent of p, has zero propagation and decomposes as Fp(0) =W0⊕ idL2(0,∞),
where W0 is a coisometry on L2(−∞, 0) (i.e., W0W
∗
0 = idL2(−∞,0)) of index one,
• and similarily Fp(1) is independent of p, has zero propagation and decomposes as Fp(1) =
W1 ⊕ idL2(−∞,1) for an index one coisometry W1 on L2(1,∞).
We can now adapt the constructions of [20, Sections 5] to our needs. Proofs of our claims can
be found in that reference. Given ε > 0 we can find a partition of unity {σε,i}i∈N on X such that
• each σε,i is continuous, Γ-invariant and Γ-cocompactly supported,
• σε,iσε,j = 0 if |i− j| ≥ 2,
• each σε,i is ε2
−i-Lipschitz.
8Alternatively one can use that topological K-theory of C∗-algebras is a special case of topological K-theory of
Banach algebras [3, Chapters 5 and 8]. Since Ψ will clearly be a homomorphism of Banach algebras, it will induce
a homomorphism on K-theory even without the ∗-property.
Equivariant Coarse (Co-)Homology Theories 35
It can be constructed, for example, by starting with an analogous but non-equivariant partition
of unity {sε,i}i∈N on [0,∞) and defining σε,i(x) := sε,i(dist(x,Γ · x0)) for some fixed x0 ∈ X.
For each n ∈ N \ {0} one can now find εn > 0 and pn,i ∈ (0, 1] such that the Γ-equivariant
operators
Gn(s) :=
∞∑
i=1
ρ(σεn,i)⊗ Fpn,i(s) ∈ B
(
H ⊗ L2(R)
)
have propagation at most 1
n for all s ∈ [0, 1]. Note that these operators are all bounded by
2 supp∈(0,1] ∥Fp∥, because the second assumption on the partition of unity implies
∥G(v)∥ ≤
∥∥∥∥∑
i even
(ρ(σεn,i)⊗ Fpn,i(s))v
∥∥∥∥+ ∥∥∥∥∑
i odd
(ρ(σεn,i)⊗ Fpn,i(s))v
∥∥∥∥
≤
(∑
i even
∥ idH ⊗Fpn,i∥2 · ∥(ρ(σεn,i)⊗ idL2(R))v∥2
)1/2
+
(∑
i odd
∥ idH ⊗Fpn,i∥2 · ∥(ρ(σεn,i)⊗ idL2(R))v∥2
)1/2
≤ 2 sup
p∈(0,1]
∥Fp∥ · ∥v∥
for all v ∈ H ⊗ L2(R). Furthermore, these operators are equicontinuous in s because of the
equicontinuity of the Fp. By the affine linear interpolation Gt := (t− n)Gn+1 + (n+ 1− t)Gn for
t ∈ [n, n+ 1] we extend these maps to a bounded continuous map
G : [1,∞)→ C[0, 1]⊗B
(
H ⊗ L2(R)
)
, t 7→ Gt
and define the bounded continuous map
R : [1,∞)→ C[0, 1]⊗B
((
H ⊗ L2(R)
)2)
,
t 7→ Rt :=
(
2Gt −GtG
∗
tGt GtG
∗
t − idH⊗L2(R)
idH⊗L2(R)−G∗
tGt G∗
t
)
.
Each Rt is invertible with inverse
R−1
t =
(
G∗
t idH⊗L2(R)−G∗
tGt
GtG
∗
t − idH⊗L2(R) 2Gt −GtG
∗
tGt
)
and we define the idempotents Q := idH⊗L2(R)⊕0 and Pt := RtQR
−1
t in C[0, 1] ⊗ B
((
H ⊗
L2(R)
)2)
.
The proof of [20, Lemma 5.11] now shows the following properties of these idempotents:
• The operators Q ⊗ idD, Pt(s) ⊗ idD for s ∈ [0, 1], t ∈ [1,∞) and T ⊗ id(L2(R))2 for T ∈
C∗
Γ(C
∗(X;D)) are all equivariant and have finite propagation. Thus, they are multipliers
of C∗
Γ(C
∗(X × [0, 1];D)).
• For all s ∈ [0, 1], t ∈ [1,∞) and T ∈ C∗
Γ(C
∗(X;D)) we have
((Q− Pt(s))⊗ idD) ◦ (T ⊗ id(L2(R))2) ∈ C∗
Γ(C
∗(X × [0, 1];D)).
The above-mentioned proof goes directly through for t ∈ N \ {0}. For all other t we
additonally need to exploit the second of the above-mentioned properties of the Fp, i.e.,
that for fixed s and varying p the Fp(s) are compact perturbations of one another.
36 C. Wulff
• For all e ∈ C
(
S1
)
⊗ C∗
Γ(C
∗(X;D)) the commutators[
Pt ⊗ idC(S1)⊗D, e⊗ id(L2(R))2
]
converge in norm to zero as t→∞.
These properties imply that algebra homomorphism
Ψ: C
(
S1
)
⊗ C∗
Γ(X;D)→ A
(
C[0, 1]⊗ C
(
S1
)
⊗D
(
C∗
Γ(X × [0, 1];D)
))
,
which is the non-relative version of (4.5), can now be defined by mapping e ∈ C
(
S1
)
⊗
C∗
Γ(C
∗(X;D)) to the class represented by the function
t 7→
(
s 7→ ((Pt(s)⊗ idD) ◦ (e(s)⊗ id(L2(R))2), (e(s)⊗ idL2(R))⊕ 0)︸ ︷︷ ︸
∈D
(
C∗
Γ(X×[0,1];D))
)
)
.
It is not a ∗-homomorphism, because the Rt are not unitaries and hence the Pt are not self-adjoint.
To finish the proof of the absolute case, it remains to analyze what happens at s = 0 and
s = 1. At s = 0 we have Gt(0) = idH ⊗Fp(0) = (idH ⊗W0) ⊕ idH⊗L2(0,∞) for all t ∈ [1,∞),
p ∈ (0, 1] and a calculation yields
Pt(0) =
(
idH⊗L2(−∞,0) 0
0 idH ⊗(idL2(−∞,0)−W ∗
0W0)
)
⊕ id(H⊗L2(0,∞))2
= Q⊕ (idH⊗D ⊗p0)
for a rank one orthogonal projection p0 ∈ B(L2(R)) which is supported over 0 ∈ [0, 1] with
respect to ρ′. It is now straightforward to verify that Θ ◦ (ev0)∗ ◦Ψ∗ is the same map as the one
induced by the ∗-homomorphism C∗
Γ(X,A;D)→ C∗
Γ((X,A)× [0, 1];D), T 7→ T ⊗ p0, and this is
exactly (e0)∗. Analogously we obtain Θ ◦ (ev1)∗ ◦Ψ∗ = (e1)∗.
Since all Pt(s) have propagation bounded by one, the constructions in the last two paragraphs
are compatible with taking the quotients by the ideals C∗
Γ(A ⊂ X;D), C∗
Γ(A×[0, 1] ⊂ X×[0, 1];D)
and then the relative case follows immediately. ■
We summarize:
Theorem 4.22. Let Γ be a discrete group. The K-theory groups of the Γ-equivariant relative
Roe algebras with coefficients in a possibly graded Γ-C∗-algebra constitute a Γ-equivariant coarse
homology theory on the admissible category of pairs of proper metric isometric Γ-spaces which
satisfies the flasqueness and the strong homotopy axiom.
The Roe algebra can also be modified as follows such that the resulting coarse homology
theory satisfies the coronality instead of the flasqueness axiom. The compact operators in
C∗
Γ(A ⊂ X;D) form an ideal KΓ(A ⊂ X;D) := C∗
Γ(A ⊂ X;D) ∩ K(H) which is 0 if A = ∅ and
equal to the C∗-algebra K(H)Γ of Γ-equivariant compact operators on H if A ̸= ∅ and X is
coarsely connected, but KΓ(A ⊂ X;D) ⫋ K(H)Γ if X is coarsely disconnected. We also define
KΓ(X;D) := KΓ(X ⊂ X;D) = C∗
Γ(X;D) ∩ K(H) ⊂ C∗
Γ(X;D). We can now define the coronal
relative Γ-equivariant Roe algebras with coefficients in D as
C∗
/K;Γ(X,A;D) :=
C∗
Γ(X;D)/KΓ(X;D)
C∗
Γ(A ⊂ X;D)/KΓ(A ⊂ X;D)
∼=
C∗
Γ(X;D)
C∗
Γ(A ⊂ X;D) + KΓ(X;D)
and we note that C∗
/K;Γ(X,∅;D) = C∗
Γ(X;D)/KΓ(X;D). These C∗-algebras vanish if X is
relatively compact, because in this case C∗
Γ(X;D) = KΓ(X;D). Note also that we have
C∗
/K;Γ(X,A;D) ∼= C∗
Γ(X,A;D) if A contains points from all coarse components of X.
Equivariant Coarse (Co-)Homology Theories 37
Theorem 4.23. The functor K∗(C
∗
/K;Γ(−,−;D)) is a Γ-equivariant coarse homology theory on
the admissible category of pairs of proper metric isometric Γ-spaces satisfying the coronality and
the strong homotopy axiom.
Proof. Functoriality and independence of F (−,−) := K∗(C
∗
/K;Γ(−,−;D)) from the chosen
representation follows just as for G(−,−) := K∗(C
∗
Γ(−,−;D)), because the constructions on the
level of C∗-algebras pass to the quotients. We know that the excision isomorphism for G holds
already at the level of C∗-algebras and from here one readily deduces that the same is true for F .
The long exact sequence and coronality are clear from the construction. Finally, strong homotopy
invariance can be proven just like Lemma 4.21 simply by taking another quotient by the ideal
compact operators in the last part of the proof. ■
5 From topological to coarse (co-)homology theories
Let Γ be a fixed discrete group and let CGBG2
Γ be the category of pairs of countably genera-
ted proper isocoarse Γ-spaces (see Definition 2.21) of bornologically bounded geometry (see
Definition 2.4) and Γ-equivariant coarse maps. It is an admissible category in the sense of
Definition 3.1.
The purpose of this section is to construct coarse (co-)homology theories on CGBG2
Γ from
generalized (co-)homology theories on so-called σ-locally compact Γ-spaces by means of a Rips-
complex construction. Coarse (co-)homology theories constructed in this way always satisfy the
strong homotopy axiom. For a very useful special type of (co-)homology theories, the so-called
single-space (co-)homology theories which also satisfy the so-called strong excision axiom, the
resulting coarse (co-)homology theories also satisfy the flasqueness axiom and can be related to
the (co-)homology theories of certain coronas via the so-called transgression maps.
The idea of constructing coarse (co-)homology theories in this way goes back to the definition
of coarse K-theory in [10] and was generalized to non-equivariant single-space cohomology theories
in [13, Section 4]. New in our present discussion are the group actions and that we take a much
deeper look at the details.
A different approach to constructing coarse homology theories is via anti-Čech systems, see
[21, Section 2], [27, Definition 3.5] and [37, Section 5.5]. Unfortunately, it seems to be unsuitable
for cohomology theories, because there instead of simply taking the direct limits one has to find
groups satisfying lim−→
1-sequences, cf. [35, Chapter 3].
5.1 Categories of σ-spaces
In this subsection we also allow arbitrary locally compact groups Γ, but we will return to discrete
ones in the following two subsections.
Definition 5.1 (cf. [10, Section 2], [43, Definition 3.1]). Let Γ be a locally compact group. A σ-
locally compact Γ-space X is an increasing sequence X0 ⊂ X1 ⊂ X2 ⊂ X3 ⊂ · · · of locally compact
Hausdorff spaces equipped with continuous Γ-actions such that for all m ≤ n the space Xm is
closed in Xn and carries the subspace topology and the restricted Γ-action. We call X a σ-compact
Γ-space, if in addition all Xn are compact.9
By an abuse of notation we will use the symbol X for the set X =
⋃
n∈NXn endowed with
the induced action and the final topology, i.e., a subset A ⊂ X is open/closed if and only if
every intersection An := A∩Xn is open/closed. The final topology has the property that a map
from X into another topological space is continuous if and only if it is continuous on every Xn.
9Note that this is not the same as what is usually known as σ-compact spaces in the literature.
38 C. Wulff
Note also that the final topology on X is Hausdorff itself: If x, y ∈ X are two distinct points,
say x, y ∈ Xn, then there is a function f ∈ C0(X) with f(x) = 0 and f(y) = 1. This function can
be extended inductively to all Xm with m ≥ n, because the one-point compactification X+
m is
a closed subspace of the normal Hausdorff space X+
m+1. Thus, one obtains a continuous extension
F : X → C with F (x) = 0 and F (y) = 1.
Any locally compact Hausdorff Γ-space X can be considered also as a σ-locally compact
Γ-space by assigning to it the constant sequence Xn := X.
Definition 5.2. We say that a σ-locally compact Γ-space X =
⋃
n∈NXn is a subspace of
Y =
⋃
n∈N Yn if X ⊂ Y is Γ-invariant, X ∩ Yn = Xn and each Xn carries the subspace topology
of Yn and the restricted Γ-action.
Definition 5.3. Let X , Y be σ-locally compact Γ-spaces given by the filtrations (Xm)m∈N
and (Yn)n∈N, respectively, and let f : X → Y be a map.
• We call f a σ-map if for every m ∈ N there exists n ∈ N such that f(Xm) ⊂ Yn. Note
that such an f is continuous in the final topologies if and only if all the restricted maps
f |Xm : Xm → Yn are continuous.
• We call a continuous σ-map f proper if the preimages of all σ-compact subspaces of Y
under f are σ-compact subspaces of X , or equivalently, if the restricted maps f |Xm :
Xm → Yn are proper continuous maps.
Using these notions we can introduce the categories of spaces on which we will consider
(co-)homology theories.
Definition 5.4.
• Let σCH2
Γ be the category whose objects are pairs (X ,A) of σ-compact Γ-spaces with
A ⊂ X a closed subspace10 and whose morphisms between (X ,A) and (Y,B) are continuous
Γ-equivariant σ-maps X → Y which restrict to maps A → B.
• Let σLCH2
Γ be the category whose objects are pairs (X ,A) of σ-locally compact Γ-spaces
with A ⊂ X a closed subspace and whose morphisms between (X ,A) and (Y,B) are proper
continuous Γ-equivariant σ-maps X → Y which restrict to maps A → B.
• The category σLCH+
Γ has objects the σ-locally compact Γ-spaces and its morphisms
between X and Y are proper continuous Γ-equivariant σ-maps U → Y , where U ⊂ X is an
open subspace. The composition of two such morphisms X ⊃ U f−→ Y and Y ⊃ V g−→ Z is
defined to be the morphism X ⊃ f−1(V)
g◦f |f−1(V)−−−−−−−→ Z.
• The category σLCH+,2
Γ is a mixture of the last two: The objects (X ,A) are the same as
in σLCH2
Γ and the morphisms are of the form X ⊃ U → Y as in σLCH+
Γ but with the
additional property that they map A ∩ U into B. If this is the case, then the restriction
A ∩ U → B is also morphisms in σLCH+
Γ .
• The categories CH2
Γ, LCH
2
Γ, LCH
+,2
Γ and LCH+
Γ are the restrictions of the above categories
to pairs of compact Hausdorff spaces and locally compact Hausdorff spaces, respectively.
Equivalently, one can define them by removing all σ’s in the above four parts of the
definition. Everything that will be said in the remainder of this section about the above
four categories is also true in complete analogy for these non-σ-counterparts and therefore
we shall not mention the corresponding statements explicitly.
• If the index Γ is omitted from the notation, this means we do not consider group actions,
i.e., Γ = 1.
10Recall that subspaces are always Γ-invariant by Definition 5.2.
Equivariant Coarse (Co-)Homology Theories 39
The definition of the morphisms in the category σLCH+
Γ may appear a bit strange at first
sight, but note that they are exactly in one-to-one correspondence with the basepoint-preserving
continuous Γ-equivariant σ-maps between the one-point σ-compactifications
X+ :=
⋃
n∈N
X+
n =
⋃
n∈N
Xn ∪ {∞},
i.e., the σ-compact spaces defined by the corresponding sequence of one-point compactifications.
In other words, the functor σLCH+
Γ → σCH2
Γ which maps a space X to the pair of spaces
(X+, {∞}) and a morphism X ⊃ U f−→ Y to the continuous σ-map
f+ : (X+,∞)→ (Y+,∞), x 7→
{
f(x), x ∈ U ,
∞, x ∈ X+ \ U
is a full and faithful functor.
Conversely, a right inverse to the above functor is the composition of the full and faithful
inclusion of categories σCH2
Γ → σLCH2
Γ with the faithful inclusion of categories σLCH2
Γ →
σLCH+,2
Γ and the functor σLCH+,2
Γ → σLCH+
Γ which maps pairs of spaces (X ,A) to their
difference X \ A and morphisms (X ,A)→ (Y,B) which are given by the data X ⊃ U f−→ Y to
the morphism X \ A ⊃ f−1(Y \ B)
f |f−1(Y\B)−−−−−−−→ Y \ B.
Remark 5.5. The most convenient way to construct a morphism U → V in the category σLCH+
Γ
is by providing a morphism (X ,A) → (Y,B) with X \ A = U and Y \ B = V in the category
σLCH2
Γ and applying the above construction, i.e., applying the last two of those three functors.
Now that we have explained our categories of spaces, we can go on to product spaces and
homotopies.
Definition 5.6. Let X be a σ-locally compact Γ-space and Y a σ-locally compact Γ′-spaces
given by the filtrations (Xm)m∈N and (Yn)n∈N, respectively. Their product X × Y is the σ-
locally compact Γ× Γ′-space defined by the sequence of locally compact Hausdorff Γ× Γ′-spaces
(Xn × Yn)n∈N.
Again, this is not a cartesian product in the category of σ-locally compact Hausdorff spaces.
Note that the union
⋃
n∈NXn × Yn equipped with the final topology is homeomorphic to
the product of the spaces
⋃
n∈NXn and
⋃
n∈N Yn and hence the interpretation of the expression
X × Y as a topological space is unambiguous.
The following two lemmas and the corollary are obvious. The first lemma says that the cross
product is functorial in both variables, the second treats inclusions of one of the factors as slices
and projections onto one of the factors, and the corollary explains adequate notions of homotopy.
Lemma 5.7. The assignment(
(X ,A), (Y,B)
)
7→ (X × Y,A× Y ∪ X × B)
gives rise to functors
σCH2
Γ × σCH2
Γ′ → σCH2
Γ×Γ′ ,
σLCH2
Γ × σLCH2
Γ′ → σLCH2
Γ×Γ′ ,
σLCH+,2
Γ × σLCH+,2
Γ′ → σLCH+,2
Γ×Γ′
and the assignment (X ,Y) 7→ X × Y gives rise to a functor
σLCH+
Γ × σLCH+
Γ′ → σLCH+
Γ×Γ′ .
40 C. Wulff
In each case, the morphisms are mapped in the obvious way. They are compatible with each
other in the sense that they commute with the three functors mentioned in the paragraph before
Remark 5.5.
Lemma 5.8. Fixing (Y,B) (or only Y) in the above lemma gives rise to functors
−× (Y,B) : σCH2
Γ → σCH2
Γ×Γ′ ,
−× (Y,B) : σLCH2
Γ → σLCH2
Γ×Γ′ ,
−× (Y,B) : σLCH+,2
Γ → σLCH+,2
Γ×Γ′ ,
−× Y : σLCH+
Γ → σLCH+
Γ×Γ′
and for each y ∈ Y the maps X → X ×{y} ⊂ X ×Y are proper continuous σ-maps and constitute
a natural transformation from the respective identity functors to each of the above functors.
Conversely, if Y is σ-compact and B = ∅, then the projection maps X × Y → X are proper
continuous σ-maps and constitute a natural transformation from −× Y to the identity functors
in each of these categories and this natural transformation is left inverse to the abovementioned
ones.
Corollary 5.9. For Y = I := [0, 1] the unit interval with trivial Γ′ = 1-action and B = ∅, the
above natural tranformations give rise to an adequate notion of homotopy in each of the four
categories σCH2
Γ, σLCH
2
Γ, σLCH
+,2
Γ and σLCH+
Γ .
5.2 Generalized (co-)homology theories and the Rips complex
From all of the categories of spaces that we have introduced in the last section, CH2
Γ, LCH
2
Γ,
σCH2
Γ and σLCH2
Γ are almost admissible categories in the sense of Eilenberg and Steenrod [9,
Section I.1], except for the slight difference that the objects are not just topological spaces but
contain a little bit of additional information in the form of the group action and possibly the
filtration by closed subspaces.
Definition 5.10. By a (co-)homology theory (or Γ-equivariant (co-)homology theory) on CH2
Γ,
LCH2
Γ, σCH
2
Γ or σLCH2
Γ we understand a collection of covariant (contravariant) functors from this
category to the category of Z-graded abelian groups together with natural transformations, the
(co-)boundary maps, just as in the axiomatic set-up by Eilenberg and Steenrod, which satisfies
the homotopy, exactness and excision axioms, but not necessarily the dimension or the additivity
axioms, cf. [9, Sections I.3 and I.3c].
In the following, we will always be interested in such a homology theory EΓ
∗ or such a coho-
mology theory E∗
Γ on the category σLCH2
Γ. The purpose of this section is to show that applying
them to the Rips complex yields coarse (co-)homology theories on CGBG2
Γ which satisfy the
strong homotopy axiom. From now on, Γ is a discrete group.
We first construct the Rips complex for locally finite countably generated proper isocoarse
Γ-spaces. If E is a Γ-invariant entourage, then we define PE(X) to be the geometric realization
of the simplicial complex with vertex set X and with x0, . . . , xn ∈ X spanning an n-simplex
iff (xi, xj) ∈ E for all 0 ≤ i, j ≤ n. Note that we could equally well consider only symmetric
entourages for this construction, because PE(X) = PE∩E−1(X), but there is no need at all for
doing so. The space PE(X) is locally compact, because X is locally finite, and it is a proper
Γ-space, because E is Γ-invariant and the action of Γ on X is proper.
Let S := {En | n ∈ N} be a countable generating set of the coarse structure. We may
assume ∆X = E0 ⊂ E1 ⊂ · · · and that each entourage E is contained in one of the En, because
otherwise we simply replace En by the finite union of all the entourages that can be obtained
from E1, . . . , En by up to n operations of the form (3)–(5) of Definition 2.1. Furthermore, we
may assume that all En are Γ-equivariant, because the Γ-action is isocoarse.
Equivariant Coarse (Co-)Homology Theories 41
Definition 5.11. The Rips complex of X with respect to the generating set S is the σ-locally
compact Γ-space P(X,S) :=
⋃
m∈N Pm(X), where we have abbreviated Pm(X) := PEm(X).
Note that as a topological space, the Rips complex is simply the full simplicial complex of X,
but as a σ-space it depends on the choice of the generating set S, because the sequence of the
Pm(X) is an essential part of the data. Nevertheless, the following lemma justifies to drop the
generating set from the notation and simply write P(X).
Lemma 5.12. For two different choices of generating sets S, S′ as above, the resulting Rips
complexes are canonically isomorphic in the category σLCHΓ of σ-locally compact Γ-spaces and
proper continuous Γ-equivariant maps via the identity map id : P(X,S)→ P(X,S′).
Proof. If S′ is given by the sequence ∆X = E′
0 ⊂ E′
1 ⊂ · · · , then for every m ∈ N there is n ∈ N
such that Em ⊂ E′
n and hence PEm(X) ⊂ P ′
En
(X). Therefore the identity is a σ-map. ■
Given in addition a Γ-invariant subspace A ⊂ X, its Rips complex P(A) with respect to the
restricted generating set {E ∩ (A×A) | E ∈ S′} is a closed subspace of P(X). If f : (X,A)→
(Y,B) is a Γ-equivariant coarse map between two pairs of locally finite countably generated
coarse Γ-spaces, then affine linear extension defines a proper continuous Γ-invariant σ-map
P(f) : (P(X),P(A)) → (P(Y ),P(B)), i.e., a morphism in σLCH2
Γ, the subset PE(X) being
mapped to PF (Y ) if (f × f)(E) ⊂ F . It is readily verified that close maps X → Y induce
homotopic maps P(X) → P(Y ), where the homotopy is affine linear, but we even have the
following much stronger result.
Lemma 5.13. Let f, g : (X,A) → (Y,B) be Γ-equivariant coarse maps between two pairs of
locally finite countably generated proper isocoarse Γ-spaces. If f , g are Γ-equivariantly gene-
ralized coarsely homotopic, then their induced maps between the Rips complexes P(f), P(g):
(P(X),P(A))→ (P(Y ),P(B)) are Γ-equivariantly homotopic.
Proof. We recall the choices that we have already made at the beginning of the proof of
Lemma 4.4. Let H : (X × [a, b], EU )→ (Y, EY ) be a Γ-equivariant generalized coarse homotopy
between f and g which maps A× [a, b] into B. For each point x ∈ X we choose a = sx,0 < sx,1 <
· · · < sx,kx = b such that (s, sx,j) ∈ Ux and (s, sx,j+1) ∈ Ux for all s ∈ [sx,j , sx,j+1], j = 0, . . . , kx−1
and we can assume that sx,j = sxγ,j for all γ ∈ Γ, x ∈ X and j = 0, . . . , kx− 1 = kxγ − 1. Now, if
x0, . . . , xn span an n-simplex in PE(X) for some entourage E ∈ EX and if s ∈
⋂n
i=0[sxi,ji , sxi,ji+1],
then the points
(x0, sx0,j0), . . . , (xn, sxn,jn), (x0, sx0,j0+1), . . . , (xn, sxn,jn+1)
span an 2n-simplex in P(E×̄∆[a,b])◦EU (X). Due to this property the map H̃ : P(X)× [a, b]→ P(Y )
we are now about to define will obviously be a σ-map.
We start the construction of H̃ by demanding that H̃ shall agree with H on the subset
{(x, sx,j) | x ∈ X ∧ j = 0, . . . , kx}. Then we extend it affine linearly to a map on X × [a, b],
that is
H̃(x, (1− r)sx,j + rsx,j+1) := (1− r) ·H(x, sx,j) + r ·H(x, sx,j+1)
for all j = 0, . . . , kx − 1 and r ∈ [0, 1]. And finally, we extend it again affine linearly to all of
P(X)× [a, b] by
H̃
(∑
x∈X
λxx, s
)
:=
∑
x∈X
λxH̃(x, s).
Continuity of H̃ is obvious and topological properness of H̃ follows directly from the coarse
geometric properness of H. It is also a σ-map, because it clearly maps PE(X) into
P(H×H)((E×̄∆[a,b])◦EU )(Y ), and it obviously maps P(A) into P(B). Finally, Γ-equivariance follows
from the Γ-invariant choice of the sx,j . ■
42 C. Wulff
We are now going to apply these constructions to discretizations of pairs (X,A) of countably
generated proper isocoarse Γ-spaces of bornologically bounded geometry, i.e., objects in CGBG2
Γ.
According to Lemma 2.26 there are Γ-invariant discretizations ιX : X ′ ⊂ X and A′ ⊂ A such that
there are Γ-equivariant coarse equivalences πX : X → X ′ and πA : A→ A′ which are the identities
on X ′ and A′, respectively. By performing the obvious modifications of X ′ and πX , if necessary,
we may assume that A′ ⊂ X ′ and πA = πX |A. Then the inclusion ιX : (X ′, A′) → (X,A) is
a Γ-equivariant coarse equivalence with πX being a Γ-equivariant coarse inverse up to closeness.
Definition 5.14. Let EΓ
∗ be a generalized Γ-equivariant homology theory or E∗
Γ a generalized
Γ-equivariant cohomology theory. Then its coarsification EXΓ
∗ or EX∗
Γ is defined by the groups
EXΓ
∗ (X,A) := EΓ
∗ (P(X ′),P(A′)) or EX∗
Γ(X,A) := E∗
Γ(P(X ′),P(A′)),
respectively, where (X ′, A′) denotes some fixed discretization of (X,A) of the type described
above.
Theorem 5.15. Up to canonical isomorphism, the coarsifications EXΓ
∗ (X,A) and EX∗
Γ(X,A) are
independent of the choice of discretization. Furthermore, they are the groups of a Γ-equivariant
(co-)homology theory that satisfies the strong homotopy axiom.
Proof. If there is another pair of spaces with discretization (Y ′, B′)
πY↼−−−−⇁
ιY
(Y,B) of the same
type and if f : (X,A)→ (Y,B) is a Γ-equivariant coarse map, then the Γ-equivariant coarse map
f ′ := πY ◦ f ◦ ιX : (X ′, A′) → (Y ′, B′) is up to closeness independent of the choice of πY and
consequently the induced map on the Rips complexes P(f) : (P(X ′),P(A′))→ (P(Y ′),P(B′)) is
up to homotopy in the category σLCH2
Γ independent of the choice of πY by Lemma 5.13. Thus,
we obtain an induced group homomorphism f∗ : EX
Γ
∗ (X,A)→ EXΓ
∗ (Y,B) and f∗ : EX∗
Γ(Y,B)→
EX∗
Γ(X,A) which is independent of the choice of πY .
Also, if f1, f2 are generalized coarsely homotopic, then so are f ′1, f
′
2 and therefore P(f1),
P(f2) are homotopic in the category σLCH2
Γ, so (f1)∗ = (f2)∗ and (f1)
∗ = (f2)
∗.
If there is a third pair of spaces with discretization (Z ′, C ′)
πZ↼−−−−⇁
ιZ
(Z,C) and g : (Y,B)→ (Z,C)
is a Γ-equivariant coarse map, then g′ ◦ f ′ is close to (g ◦ f)′ and hence P(g′) ◦P(f ′) is homotopic
to P(g′ ◦ f ′), so g∗ ◦ f∗ = (g ◦ f)∗ and f∗ ◦ g∗ = (g ◦ f)∗.
Applying all of this to the identity map between the same pair of spaces (X,A) but equipped
with different discretizations (X ′, A′), we see that the groups EXΓ
∗ (X,A) and EX∗
Γ(X,A) are
independent of the choice of discretization up to canonical isomorphism.
Thus, so far we have shown that the groups form a covariant or a contravariant functor
from CGBG2
Γ to the category of abelian groups which satisfies the strong homotopy axiom. The
exactness axiom for EXΓ
∗ , EX
∗
Γ follows immediately from the exactness axiom for EΓ
∗ , E
∗
Γ.
Finally, but most complicatedly, it remains to show the excision axiom. Let (X \ C,A \ C) ⊂
(X,A) be an excision (cf. Definition 3.2). We can choose discretizations C ′ ⊂ C, A′ ⊂ A and
X ′ ⊂ X as above such that C ′ ⊂ A′ ⊂ X ′ and then (X ′ \C ′, A′ \C ′) ⊂ (X ′, A′) is an excision, too.
Let α : P(X)→ [0, 1] be the continuous Γ-invariant function which is 1 on A′, 0 on X ′ \A′ and
then extended over all simplices affine linearily. In the same way we construct β : P(X)→ [0, 1]
being 1 on C ′ and 0 on X ′ \ C ′. We define the closed subspace A := α−1
[
1
3 , 1
]
and the open
subspace C := β−1
(
2
3 , 1
]
of P(X ′). Note that the closure of C is contained in the interior of A,
because β ≤ α, and therefore the inclusions (P(X ′) \ C,A \ C) → (P(X ′),A) are topological
excisions and thus induce isomorphisms on (co-)homology.
The proof can now be finished by showing that the subspaces P(A′) ⊂ A, P(X ′ \ C ′) ⊂
P(X ′) \ C and P(A′ \ C ′) ⊂ A \ C are deformation retracts, because then homotopy invariance
and the long exact sequences will show that the inclusions (P(X ′),P(A′)) → (P(X ′),A) and
(P(X ′ \ C ′),P(A′ \ C ′))→ (P(X ′) \ C,A \ C) induce isomorphisms on (co-)homology.
Equivariant Coarse (Co-)Homology Theories 43
Constructing a retraction A → P(A′) is straightforward: A point x ∈ A has a decomposition
x = λa+ (1− λ)b with a ∈ P(A′) and b ∈ P(X ′ \A′) in which λ ≥ 1
3 . In particular, λ ̸= 0 and
therefore a is uniquely determined. Due to this uniqueness, mapping x 7→ a gives us a continuous
Γ-equivariant retraction which is homotopic to the identity via the obvious affine linear homotopy.
Furthermore, both the retraction and the homotopy respect the filtration, because if x lies in a
simplex, then a and the whole path between x and a lie in the same simplex. Exactly the same
procedure provides us with a deformation retraction P(X ′) \ C → P(X ′ \ C ′).
The tricky part is to construct the deformation retraction A \ C → P(A′ \ C ′) and here we
will need the excisiveness condition. Let ∆X′ = E0 ⊂ E1 ⊂ · · · be a sequence of symmetric
Γ-invariant entourages of X ′ such that every entourage is contained in some En, and set E−1 = ∅.
We choose a point xO ∈ O in each Γ-orbit O of X ′, O = xOΓ, and let ΓxO denote the stabilizer
of xO. Now, there is a unique nO ∈ N such that xO ∈ PenEn(A
′ \ C ′) \ PenEn−1(A
′ \ C ′) and
therefore in particular O = xOΓ ⊂ PenEn(A
′ \C ′). Choose yO ∈ A′ \C ′ such that (xO, yO) ∈ En.
Note that for all γ ∈ Γ, γ′ ∈ ΓxO we have (xOγ, yOγ
′γ) ∈ En by Γ-invariance of En. Also, for
O ⊂ X ′ \ C ′, xO ∈ X ′ \ C ′, we have yO = xO.
We can now define a map r : P(X ′) → P(A′ \ C ′) as the affine linear extension of the
Γ-equivariant extension of the mapping xO 7→
∑
γ′∈ΓxO
yOγ
′, that is
r :
∑
O⊂X′, γ∈Γ
λO,γxOγ 7→
∑
O⊂X′, γ∈Γ, γ′∈ΓxO
λO,γyOγ
′γ.
This map is clearly well defined, Γ-equivariant, continuous and restricts to the identity on
P(A′ \ C ′).
It is not a σ-map itself, but its restriction to A\C is: If x ∈ A\C, then each simplex of P(X ′)
which contains x must have at least one vertex in A′ (because α(x) ≥ 1
3
)
and one vertex in
X ′ \C
(
because β(x) ≤ 2
3
)
. Thus, if this particular simplex lies in PE(X
′), then all of its vertices
lie in PenE(A
′) ∪ PenE(X
′ \ C ′) ⊂ PenEn(A
′ \ C ′), where n ∈ N only depends on E. This shows
(A \ C) ∩ PE(X
′) ⊂ PE(PenEn(A
′ \ C ′)) and r clearly maps this subset into PEn◦E◦En(A
′ \ C ′).
The same applies to the homotopy between r and the identity on A \ C given by affine linear
interpolation. Therefore, P(A′ \ C ′) ⊂ A \ C is a deformation retract. ■
5.3 Single space (co-)homology theories and transgression maps
An important class of coarse (co-)homology theories are the coarsifications of so-called single-space
(co-)homology theories. We will show in this section that they always satisfy the flasqueness
axiom and admit the so-called transgression maps.
Definition 5.16. A (co-)homology theory on σCH2
Γ or σLCH2
Γ is called a single-space (co-)ho-
mology theory (or Γ-equivariant single space (co-)homology theory, if we want to emphasize the
group action), if it factors through σLCH+
Γ up to natural isomorphism. This property is called
the strong excision axiom. Equivalently, these (co-)homology theories can be axiomatized using
solely the absolute (co-)homology groups11 as follows:
• A Γ-equivariant single-space homology theory EΓ
∗ for σ-locally compact spaces is a collection
of covariant homotopy functors EΓ
n indexed by n ∈ Z from the category σLCH+
Γ to the
category of abelian groups together with boundary maps ∂n : E
Γ
n(X \ A)→ EΓ
n−1(A) such
that the long sequences
· · · → EΓ
n(A)→ EΓ
n(X )→ EΓ
n(X \ A)
∂n−→ EΓ
n−1(A)→ · · · (5.1)
are exact and natural in (X ,A) ∈ σLCH+,2
Γ .
11Therefore the name.
44 C. Wulff
• Dually, a Γ-equivariant single-space cohomology theory EΓ
∗ for σ-locally compact spaces is
a collection of contravariant homotopy functors En
Γ indexed by n ∈ Z from the category
σLCH+
Γ to the category of abelian groups together with coboundary maps δn : En
Γ(A) →
En+1
Γ (X \ A) such that the long sequences
· · · → En
Γ(X \ A)→ En
Γ(X )→ En
Γ(A)
δn−→ En+1
Γ (X \ A)→ · · · (5.2)
are exact and natural in (X ,A) ∈ σLCH+,2
Γ .
In the same way, Γ-equivariant single-space (co-)homology theories can be introduced on CH2
Γ,
LCH2
Γ and LCH+
Γ .
This definition has already been well-established for a long time on the categories CH2, LCH2,
LCH+. In [13, Definition 4.13] we gave σ-versions for σCH2,σLCH+ under the name “generalized
Steenrod (co-)homology theories”, where “Steenrod” was meant to express that the strong
excision axiom holds. However, we drop this naming in the present paper to avoid confusion,
because other authors use it as a qualifier for the cluster axiom, which says that homology turns
disjoint unions of spaces into products of homology groups and co-homology turns disjoint unions
into direct sums (cf. [41, Definition B.2.2]). We will not use the cluster axiom here, but it should
be noted that it might be very useful for the calculation of (co-)homology of the Rips complexes
by cellular methods.
For the remainder of this section, EΓ
∗ will always denote a Γ-equivariant single-space homology
theory and E∗
Γ will always denote a generalized Γ-equivariant single-space cohomology theory for
σ-locally compact spaces and their coarsifications will be denoted by EXΓ
∗ and EX∗
Γ, respectively,
as before.
Lemma 5.17. Coarsifications of single-space (co-)homology theories satisfy the flasqueness
axiom.
Proof. Let X be a flasque space, witnessed by ϕ : X → X, and let ι : X ′ ⊂ X be a discretization
as in Lemma 2.26 with Γ-equivariant coarse equivalence π : X → X ′. Then we define the map
(P(X ′))+ × [0,∞]→ (P(X ′))+ as being equal to P(π ◦ ϕn ◦ ιX) on P(X ′)× {n}, interpolating
affine linearily inbetween and mapping everything else (i.e., {∞} × [0,∞] ∪ P(X ′) ∪ {∞}) to ∞.
From the assumptions on ϕ, it is straightforward to see that this is a Γ-equivariant proper
continous σ-map. Furthermore, it is readily checked that it descends to a homotopy in σLCH+
Γ
which shows that P(X ′) is homotopy equivalent to the empty set. The long exact sequence shows
that (co-)homology vanishes on the empty set, and therefore its coarsification vanishes on X. ■
Given a coarse space X, we let Bh(X) denote the commutative C∗-algebra of all bounded
functionsX → C of vanishing variation and B0(X) := B0(X;C) the ideal of all functions vanishing
at infinity. The Higson corona ∂hX is by definition the maximal ideal space of the quotient
C∗-algebra Bh(X)/B0(X), i.e., the compact Hausdorff space with C(∂hX) ∼= Bh(X)/B0(X).
Note that Bh(X) and B0(X) are clearly contravariantly functorial and hence ∂hX is covariantly
functorial under closeness classes of coarse maps. In particular, coarsely equivalent spaces have
homeomorphic Higson coronas. We call a compact Hausdorff space ∂X a Higson dominated
corona for the coarse space X if there is a continuous surjective map ∂hX ↠ ∂X.
If X is moreover a topological coarse space, then Bh(X) contains the sub-C∗-algebra of contin-
uous functions Ch(X) = Bh(X) ∩C(X) which contains C0(X). Its maximal ideal space is a com-
pactification X
h
of X, called the Higson compactification. We have C(∂hX) ∼= Bh(X)/B0(X) ∼=
Ch(X)/C0(X) ∼= C
(
X
h \X
)
, where the proof of the second isomorphism works exactly as the
proof of (4.2), see [10, proof of Proposition 3.7]. Thus, the Higson corona is the boundary of this
compactification. A compactification X of X with boundary ∂X is called Higson dominated, if
Equivariant Coarse (Co-)Homology Theories 45
there is a continuous surjection X
h
↠ X which restricts to the identity on X, or equivalently, if
the closure of each entourage E ⊂ X ×X intersects X × ∂X ∪ ∂X ×X only within ∂X × ∂X.
In this case ∂X := X \X is a Higson dominated corona.
If X comes equipped with a coarse Γ-action, then ∂hX and X
h
are Γ-spaces. In this case
we will always assume that Higson dominated coronas and compactifications ∂X,X are also
Γ-spaces and that the surjections are Γ-equivariant.
Now assume that X is a countably generated proper isocoarse Γ-space of bornologically
bounded geometry and ∂X a Higson dominated corona. Let X ′ ⊂ X be a Γ-equivariant
discretization as in Lemma 2.26. For each Γ-invariant entourage E of X ′, the Rips complex
PE(X
′) becomes a topological isocoarse Γ-space if we additionally equip it with the canonical
coarse structure characterized by the properties that the inclusion X ′ ⊂ PE(X
′) is a coarse
equivalence and the simplices are uniformly bounded. Then there is a Higson dominated
compactification of PE(X
′) with the same corona ∂X, which can be constructed as follows: Note
that the corona ∂X corresponds to a unital sub-Γ-C∗-algebra of C(∂hX) ∼= Bh(X)/B0(X) ∼=
Bh(X
′)/B0(X
′) ∼= Bh(PE(X
′))/B0(PE(X
′)) ∼= Ch(PE(X
′))/C0(PE(X
′)), that is, it is of the form
C/C0(PE(X
′)) for a unital sub-Γ-C∗-algebra C ⊂ Ch(PE(X
′)) and the maximal ideal space of C
is a Higson-dominated compactification PE(X ′) of PE(X
′) with boundary ∂X. Applying it to
the sequence of Rips complexes PE0(X
′) ⊂ PE1(X
′) ⊂ · · · we obtain a sequence of compact
Hausdorff Γ-spaces PE0(X
′) ⊂ PE1(X
′) ⊂ · · · , where the inclusions are the identity on ∂X. Thus,
this sequence is a σ-compactification P(X ′) of P(X ′) with boundary ∂X.
Assume furthermore that A ⊂ X is a coarse Γ-invariant subspace. Then the image of C(∂X)
under the restriction map C(∂hX) ∼= Bh(X)/B0(X)→ Bh(A)/B0(A) ∼= C(∂hA) corresponds to
a Higson dominated corona ∂A of A. By construction we have a surjection C(∂X) → C(∂A),
so ∂A can be considered a subspace of ∂X. If X ′ ⊂ X and A′ ⊂ A are two discretizations
as in Lemma 2.26 with A′ ⊂ X ′, then we obtain corresponding compactifications PE(A′) with
boundary ∂A for each entourage E of X. The function algebras C(PE(A′)) are exactly the images
of C(PE(X ′)) under the restriction maps Ch(PE(X
′))→ Ch(PE(A
′)), so we can consider PE(A′)
as a subspace of PE(X ′). Thus, the σ-compactification P(A′) is also a subspace of P(X ′) and
we note that P(A′) ∩ ∂X = ∂A.
In the proof of Theorem 5.15 we have seen that different choices of discretizations (X ′′, A′′)
lead to homotopy equivalences (P(X ′),P(A′)) ≃ (P(X ′′),P(A′′)) in the category σLCH2
Γ which
are canonical up to homotopy. They clearly extend to homotopy equivalences (P(X ′),P(A′)) ≃
(P(X ′′),P(A′′)) in the category σCH2
Γ which are the identity on the boundaries (∂X, ∂A) and
are also canonical up to homotopy. Therefore, the next definition is independent of the choice of
discretizations.
Definition 5.18. Let (X,A) be a pair of countably generated proper isocoarse Γ-spaces of
bornologically bounded geometry, ∂X a Higson dominated corona and ∂A the corresponding
corona of A. The transgression maps are the connecting homomorphisms
EXΓ
∗+1(X,A)→ EΓ
∗ (∂X, ∂A) and E∗
Γ(∂X, ∂A)→ EX∗+1
Γ (X,A)
associated to the pair of σ-locally compact spaces
(
P(X ′) \ P(A′), ∂X \ ∂A
)
.
Important for applications (see, e.g., Theorems 6.6 and 6.12 below) is the question when
transgression maps are isomorphisms. The long exact sequences and homotopy invariance
immediately imply the following result.
Lemma 5.19. If P(A′) is a deformation retract of P(X ′) in the category σCHΓ, then the
transgression maps are isomorphisms.
46 C. Wulff
5.4 Reduced (co-)homology and reduced transgression
In the absolute case A = ∅, there is a similar result to Lemma 5.19 if we consider a reduced
version of the transgression maps. To define them, we first introduce reduced (co-)homology for
σ-locally compact Γ-spaces X equipped with a decomposition X =
∐
i∈π Ki as a coproduct in the
category of σ-locally compact spaces (indexed by an arbitrary set π) with the following properties:
First, we assume that each Ki is σ-compact. And second, the subsets Ki do not have to be closed
under the Γ-action, but we do assume that each group element maps each Ki homeomorphically
onto some Kj . In case of the trivial decomposition into a single σ-compact subset K = X and
without group action we will recover the usual definition of reduced (co-)homology.
For each nonempty σ-compact space K, let CK := K × [0, 1]/K × {1} be the closed cone and
OK := K × (0, 1]/K × {1} the open cone over K. Furthermore we define C∅ and O∅ to be
the one-point space. Thanks to σ-compactness of K, the closed cone is again σ-compact and
the open cone is σ-locally compact. Then we define the closed and open multicones of X as
C+X :=
∐
i∈π CKi and O+X :=
∐
i∈πOKi, respectively. They are canonically σ-locally compact
Γ-spaces.
Definition 5.20. The reduced (co-)homology groups of a decomposed σ-locally compact Γ-
space X as above are defined as
ẼΓ
∗ (X ) := EΓ
∗+1(O+X ) and Ẽ∗
Γ(X ) := E∗+1
Γ (O+X ).
To understand this definition, we furthermore define the suspension of a σ-locally compact
Γ-space X as ΣX := X × (0, 1). The obvious deformation retraction of X × [0, 1] to X × {1}
induces a homotopy equivalence in σLCH+
Γ between X × [0, 1) and the empty space and hence
all single space (co-)homology theories vanish on them.
Definition 5.21. The suspension isomorphisms are the connecting homomorphisms
EΓ
∗+1(ΣX ) ∼= EΓ
∗ (X ) and E∗
Γ(X ) ∼= E∗+1
Γ (ΣX )
in the long exact sequences associated to the pairs of spaces (X × [0, 1),X × {0}).
Now, we can canonically consider π itself as a discrete Γ-space. As such, it is embedded in the
multicones C+X , O+X as the subspace of apices. As C+X is contractible onto π and by using
the suspension isomorphisms, the long exact sequences associated to the pair (C+X ,X × {0})
become
· · · → EΓ
∗+1(π)→ ẼΓ
∗ (X )→ EΓ
∗ (X )→ EΓ
∗ (π)→ · · · ,
· · · → E∗
Γ(π)→ E∗
Γ(X )→ Ẽ∗
Γ(X )→ E∗+1
Γ (π)→ · · · . (5.3)
These sequences imply in particular that reduced (co-)homology vanishes if the canonical Γ-
equivariant continuous σ-map X → π is a homotopy equivalence.
The connection to the well-known definition of reduced (co-)homology is now as follows: If X
contains a Γ-invariant subset which meets each Ki in exactly one point, then the map X → π
has a right inverse in σLCHΓ and the exact sequences yield canonical isomorphisms
ẼΓ
∗ (X ) ∼= ker(EΓ
∗ (X )→ EΓ
∗ (π)) and Ẽ∗
Γ(X ) ∼= coker(E∗
Γ(π)→ E∗
Γ(X )).
In the non-equivariant case and for the trivial decomposition, such that π itself is a one-
point space, these isomorphisms are exactly the classical definitions of the reduced groups via
augmentation with a point. Analogously, for non-trivial decompositions, we recognize the idea
that an augmentation with more than one point has been performed. However, we emphasize
Equivariant Coarse (Co-)Homology Theories 47
that such a right inverse π → X need not exist in the equivariant case, so our definiton via cones
is superior.
Given a closed subspace A ⊂ X equipped with the decomposition into the subsets A ∩ Ki,
we have O+A ⊂ O+X and O+X \ O+A = Σ(X \ A). As A ∩ Ki might be empty even if Ki is
not, we point out that it is important to allow empty sets in the decomposition and to define
the cones over the empty space to be the one-point space for this to work. Then, the long exact
sequences associated to the pairs (O+X ,O+A) become
· · · → ẼΓ
∗ (A)→ ẼΓ
∗ (X )→ EΓ
∗ (X ,A)→ ẼΓ
∗−1(A)→ · · · , (5.4)
· · · → Ẽ∗−1
Γ (A)→ E∗
Γ(X ,A)→ Ẽ∗
Γ(X )→ Ẽ∗
Γ(A)→ · · · . (5.5)
Lemma 5.22. Up to signs, there is a canonical natural transformations from the reduced
homological long exact sequence (5.4) to the corresponding unreduced one (5.1) and dually from
the unreduced cohomological long exact sequence (5.2) to the corresponding reduced one (5.5).
Proof. The transformations are defined by applying naturality under the morphism (O+X ,O+A)
⊃ (ΣX ,ΣA) and the suspension isomorphisms. The only nontrivial claim is that the boundary
maps EΓ
∗+1(ΣX ,ΣA) → EΓ
∗ (ΣA) and EΓ
∗ (X ,A) → EΓ
∗−1(A) correspond to each other under
suspension (and dually for cohomology), but this is only true up to the sign −1. Its proof is
a fairly standard argument from algebraic topology, although concrete references seem to be
hard to find: First, one rewrites the boundary map using suspension as
EΓ
∗ (X ,A) ∼= EΓ
∗ (X × {1} ∪A× [0, 1],A× {0})
→ EΓ
∗ (A× (0, 1],A× {0}) ∼= EΓ
∗ (ΣA)
∼= EΓ
∗−1(A)
and similarily for the boundary map for (ΣX ,ΣA). The claim is then equivalent to the fact that
the two ways of performing the double suspension EΓ
∗+1
(
Σ2A
) ∼= EΓ
∗ (ΣA)
∼= EΓ
∗−1(A) (suspending
the first copy of (0, 1) first and then the second copy versus the other way around) coincide up
to the sign −1, where the sign arises from the reflection of the square (0, 1)2 along the diagonal.
How the proof works can be seen in the follow-up article [45, Lemma 4.6.3], where a more
complicated analogous statement is shown for suspension in coarse (co-)homology. ■
Now, let X be a countably generated proper isocoarse Γ-space of bornologically bounded
geometry and X ′ ⊂ X a discretization. For each coarse component K of X, we furthermore
consider a Higson-dominated corona ∂K in a way compatible with the Γ-action. More precisely,
we assume that their coproduct ∂X :=
∐
K∈Π0(X) ∂K is a locally compact Γ-space and moreover
a quotient Γ-space of
∐
K∈Π0(X) ∂hK. We can attach each ∂K to P(X ′ ∩K) and obtain the
σ-locally compact Γ-space P(X ′) :=
∐
K∈Π0(X) P(X ′ ∩K), which contains P(X ′) as an open
and ∂X as a closed subspace. We consider all of them as equipped with the decomposition
indexed by Π0(X).12
Definition 5.23. We call a locally compact Γ-space ∂X :=
∐
K∈Π0(X) ∂K as above together
with its decomposition a componentwise Higson dominated corona of X. The associated reduced
transgression maps are the connecting homomorphisms
EXΓ
∗+1(X)→ ẼΓ
∗ (∂X) and Ẽ∗
Γ(∂X)→ EX∗+1
Γ (X)
in the long exact sequences of reduced (co-)homology associated to the pair (P(X ′), ∂X).
We now obtain the following reduced analog of Lemma 5.19.
12Note however, that the notation is slightly misleading: Here, the spaces ∂X and P(X ′) are not (σ-)compact,
if Π0(X) is not finite.
48 C. Wulff
Lemma 5.24. If each P(X ′ ∩K) is contractible and all the contractions together are compatible
with the Γ-action, then the reduced transgression maps are isomorphisms.
Proof. The hypothesis means nothing else than that the σ-locally compact Γ-space∐
K∈Π0(X) P(X ′ ∩K) is homotopy equivalent to Π0(X), canonically considered as a discrete
Γ-space. But its reduced (co-)homology groups vanish, because O+Π0(X) = (0, 1]×Π0(X) is
homotopy equivalent to the empty set. The claim now follows from the exact sequences (5.4)
and (5.5). ■
As an example, the central Theorem 5.7 of [13] says that the hypothesis is satisfied in the
non-equivariant case for the combing compactification of a proper metric space equipped with an
expanding and coherent combing.
Let us close this section by mentioning the following relation between reduced and unreduced
co-assembly.13 If X is a countably generated proper isocoarse Γ-space of bornologically bounded
geometry and K ⊂ X is a subspace whose intersection with each coarse component of X is
bounded, then we define the countably generated isocoarse Γ-space X→ := X ∪K K × N of
bornologically bounded geometry by equipping it with the obvious coarse structure. Then reduced
transgression for X→ canonically identifies with relative transgression of the pair (X→,K × N)
and with unreduced transgression for X.
6 Further examples
In the following subsections we take a closer look at coarsifications of some specific directly
constructed (co-)homology theories on σ-spaces and partly also their transgression maps.
6.1 Singular (co-)homology
Locally finite singular homology HSlf∗ (−,−;M) and compactly supported singular cohomology
HS∗c(−,−;M) with coefficients in an abelian group M are known to be (co-)homology theories
on LCH2. Let CSlf∗ (−,−,M) and CS∗c(−,−,M) denote their respective (co-)chain complexes.
If Γ is in addition a discrete group which acts on M from the left, then we can perform the
following modifications to the (co-)chain complexes to obtain (co-)homology theories on σLCH2
Γ.
First we pass to the subcomplexes CSlf,Γ∗ (−,−;M) ⊂ CSlf∗ (−,−;M) of Γ-equivariant chains and
the quotient complex CS∗c,Γ(−,−;M) := Γ\CS∗c(−,−;M), just as we did in Sections 4.1 and 4.2,
to obtain functors on LCH2
Γ. And second, if X =
⋃
n∈NXn is a σ-locally compact Γ-space with
closed subspace A =
⋃
n∈NAn, then we define
CSlf,Γ∗ (X ,A;M) := lim−→
n∈N
CSlf,Γ∗ (Xn, An;M),
CS∗c,Γ(X ,A;M) := lim←−
n∈N
CS∗c,Γ(Xn, An;M),
where all the maps in the directed systems are injective or surjective, respectively. The
usual proofs go through to show that these (co-)chain complexes define (co-)homology the-
ories HSlf∗ (−,−;M) and HS∗c(−,−;M) on σLCH2
Γ and for the sake of computations we note that
we have HSlf,Γ∗ (X ,A;M) ∼= lim−→n∈NHSlf,Γ∗ (Xn, An;M) and a Milnor-lim←−
1-sequence
0→ lim←−
n∈N
1HS∗−1
c,Γ (Xn, An;M)→ HS∗c,Γ(X ,A;M)→ lim←−
n∈N
HS∗c,Γ(Xn, An;M)→ 0.
13Compare to the end of Section 4.3.
Equivariant Coarse (Co-)Homology Theories 49
Now, if (X ′, A′) is a pair of locally finite countably generated proper isocoarse Γ-spaces,
then its ordinary Γ-equivariant coarse (co-)chain complex is nothing else but the Γ-equivariant
locally finite/compactly supported simplicial (co-)chain complex of the pair of σ-simplicial
complexes (P(X ′),P(A′)) (defined in the completely analogous way). The usual proofs using
barycentric subdivision can be adapted to the present σ-situation with Γ-actions to show that the
singular and simplicial (co-)homology of (P(X ′),P(A′)) coincide. This shows HXΓ
∗ (X
′, A′;M) ∼=
HSlf,Γ∗ (P(X ′),P(A′);M) and HX∗
Γ(X
′, A′;M) ∼= HS∗c,Γ(P(X ′),P(A′);M), that is, the ordinary
Γ-equivariant coarse (co-)homology is the coarsification of Γ-equivariant locally finite/compactly
supported singular (co-)homology.
6.2 Alexander–Spanier (co-)homology
Unfortunately, singular (co-)homology does not satisfy strong excision, so they do not provide
transgression maps. Instead one has to use other (co-)homology theories like Alexander–Spanier
cohomology and its dual.
Alexander–Spanier cohomology is known to be a single-space cohomology theory on LCH+,2
(see in particular [38, Section 6.6] or [24, Section 1.7] for a proof of the strong excision property).
By taking inverse limits of Alexander–Spanier cochain complexes one can easily generalize this
cohomology theory to a single-space cohomology theory on σLCH+,2 whose coarsification is the
(non-equivariant) ordinary coarse cohomology. As a consequence we obtain transgression maps
from the Alexander–Spanier cohomology of Higson-dominated coronas to the coarse cohomology
of a space, and one can indeed show that they agree with Roe’s original transgression map from
[35, Section 5.3]. All of this was done in [13, Section 4.4].
Care has to be taken if one tries to dualize Alexander–Spanier cohomology (or Čech cohomo-
logy) in order to obtain a homology theory which satisfies the strong excision axiom, because
most attempts of dualization destroy the long exact sequences. Massey pointed out how to
do it correctly [24, 25], providing us with a single-space homology theory on LCH+,2, which
we call Alexander–Spanier homology. It can be generalized easily to σLCH+,2 by taking direct
limits. If (X ′, A′) is a pair of uniformly locally finite countably generated coarse spaces, then
all the pairs of Rips complexes (PE(X
′), PE(A
′)) are finite dimensional and the Alexander–
Spanier homology of these pairs are isomorphic to their locally finite cellular homology by
[24, Section 4.9(4)] and hence also to their locally finite simplicial homology. Consequently,
for spaces of coarsely bounded geometry the coarsification of Alexander–Spanier homology is
(non-equivariant) ordinary coarse homology and we obtain transgression maps from ordinary
coarse homology to the Alexander–Spanier homology of Higson-dominated coronas.
The proofs of the basic properties of Alexander–Spanier (co-)homology are somewhat involved,
and therefore some non-trivial effort would probably have to be invested to see if Γ-actions can
be implemented into the definitions in the naive ways seen in Sections 4.1 and 6.2. We are not
going to pursue this question any further.
6.3 Coarse K-theory and coassembly
Given a σ-locally compact Γ-space X =
⋃
n∈NXn and a Γ-C∗-algebra D, the associated function
∗-algebra
C0(X ;D) := {f : X → D | ∀n ∈ N : f |Xn ∈ C0(Xn;D)}
equipped with the countable family of C∗-seminorms ∥f∥n := ∥f |Xn∥ and the pull-back action
γ · f : x 7→ γ · (f(xγ)) is a so-called σ-Γ-C∗-algebra, i.e., it is a complex topological ∗-algebra
whose topology is Hausdorff, generated by a countable family of Γ-equivariant C∗-seminorms and
is complete with respect to this family of seminorms. Equivalently, it is the inverse limit in the
50 C. Wulff
category of topological complex ∗-algebras of an inverse system of Γ-C∗-algebras indexed over
the natural numbers (cf. [29]).
Referring to a previous remark, the Γ-action on a σ-Γ-C∗-algebra is isometric with respect to
all the seminorms. Instead one could also pose the weaker condition that the Γ-action is only
continuous with respect to the topology on the whole algebra. The latter results in what we
would call Γ-σ-C∗-algebras, i.e., the σ and Γ are interchanged to account for the different order
in which they are implemented into the definition.
Via pullback of functions, we obtain a contravariant functor C0(−;D) : σLCH+
Γ → σC∗
Γ into the
category of σ-Γ-C∗-algebras and Γ-equivariant ∗-homomorphisms, the latter being automatically
continuous. In the case D = C this functor is actually an equivalence between the categories
(σLCH+
Γ )
op and the category of commutative σ-Γ-C∗-algebras cσC∗
Γ, i.e., a Gelfand–Naimark
type duality.
If − ⋊ Γ is any crossed product functor on the category of Γ-C∗-algebras, then we obtain
a corresponding crossed product functor from σC∗
Γ to σC∗ by applying it to an associated inverse
system of Γ-C∗-algebras and taking the inverse limit afterwards. It is a direct consequence of
[29, Proposition 5.3(2)] that the resulting crossed product functor is exact, if we started with an
exact crossed product functor. We can now extend Ktop
∗ to σ-Γ-C∗-algebras by turning (4.3) into
a definition, as was done in [11, 12].
Definition 6.1. Using Phillips’ K-theory for σ-C∗-algebras [30, 31], for any exact crossed product
functor and any σ-Γ-C∗-algebra C we define
Ktop
∗ (Γ, C) := K∗((C ⊗max P)⋊max Γ).
Furthermore, the Γ-equivariant K-theory with coefficients in a Γ-C∗-algebra D is defined as
K∗
Γ(−;D) := Ktop
−∗ (Γ,C0(−;D)).
Lemma 6.2. The functors K∗
Γ(−, D) and K−∗(C0(−;D) ⋊ Γ) for any exact crossed product
functor −⋊ Γ are Γ-equivariant single-space cohomology theories on σLCH+
Γ and they all agree
on proper σ-locally compact Γ-spaces.
Proof. The first statement follows directly from the long exact sequences and homotopy invari-
ance of Phillips’ K-theory together with continuity and exactness of the maximal tensor product
and the crossed product functors in consideration.
If X is a proper locally compact Γ-space, then C0(X;D) is a proper Γ-C∗-algebra and [8,
Remark 3.4.16] implies that all crossed products C0(X;D)⋊Γ, even the non-exact ones, coincide.
Thus, the same is true for C0(X ;D)⋊ Γ if X is a proper σ-locally compact Γ-space.
As observed in [11, Section 2.7], the Baum–Connes assembly maps yield isomorphisms
Ktop
−∗ (Γ, C) = K−∗((C ⊗max P) ⋊max Γ) ∼= K−∗(C ⋊max Γ) for all proper σ-Γ-C∗-algebras C.
Applying this to C = C0(X ;D) for proper σ-locally compact Γ-spaces X finishes the proof. ■
Definition 6.3. By the second part of the preceding lemma, the coarsifications of K∗
Γ(−, D)
and K−∗(C0(−;D) ⋊ Γ) for all exact crossed product functors − ⋊ Γ agree and we call it the
Γ-equivariant coarse K-theory KX∗
Γ(−, D).
We are also interested in the reduced K-theory groups of σ-locally compact Γ-spaces X
equipped with a decomposition X =
∐
i∈π Ki as in Section 5.4. In [12, Section 2.3], Emerson
and Meyer gave a definition thereof for undecomposed σ-compact Γ-spaces K as the K-theory
of a certain σ-Γ-C∗-algebra, which we shall call Cred(K;D). We recall and generalize their
construction and show that it agrees with the reduced groups obtained via Definition 5.21 if all
of the Ki are nonempty.
Equivariant Coarse (Co-)Homology Theories 51
Given a non-empty compact Hausdorff space K, Emerson and Meyer define Cred(K;D) as the
sub-Γ-C∗-algebra of C(K;Ms(D)) consisting of those functions for which f(x)− f(y) ∈ D ⊗ K
for all x, y ∈ K. For the empty space, we deviate and simply define Cred(∅;D) := Ms(D)
D⊗K .
If K =
⋃
n∈NKn is a σ-compact Hausdorff space, then we obtain the σ-Γ-C∗-algebra
Cred(K;D) :=
{
f : K →Ms(D) | ∀n ∈ N : f |Kn ∈ Cred(Kn;D)
}
= lim←−
n
Cred(Kn;D).
Now we generalize it to locally compact Hausdorff spaces X decomposed into compact
subsets Ki, which are allowed to be empty. To this end, for each f ∈ Cred(K;D) we let
Var(f) := supx,y∈K ∥f(x)− f(y)∥ if K ̸= ∅ and Var(f) := 0 if K = ∅. Then we define
Cred
0 (X;D) :=
{
(fi)i∈π ∈
∏
i∈π
Cred(Ki;D)
∣∣∣∣(π 7→ [0,∞), i 7→ Var(fi)
)
∈ C0(π)
}
,
recalling that we also consider π as a discrete space. Note that it can be identified as a sub-C∗-
algebra of Cb(X;Ms(D)) if all Ki are non-empty.
For the σ-locally compact space X =
⋃
n∈NXn decomposed as above into the Ki we then
define the σ-C∗-algebra
Cred
0 (X ;D) := lim←−
n∈N
Cred
0 (Xn;D),
where the locally compact subspaces Xn of the filtration of X are decomposed into the compact
subsets Xn ∩ Ki. Note that it is even a σ-Γ-C∗-algebra if X is a σ-locally compact Γ-space and
the decomposition is compatible with the Γ-action as in Section 5.4.
The purpose of this cryptic construction is to have the short exact sequence
0→ C0(X ;D)⊗ K→ Cred
0 (X ;D)→
∏
πMs(D)⊕
π(D ⊗ K)
→ 0 (6.1)
of σ-Γ-C∗-algebras.
Lemma 6.4. The reduced groups of the cohomology theories K∗(−;D), K∗
Γ(−;D)
and K−∗(C(−;D ⊗ K) ⋊ Γ) are canonically naturally isomorphic to K−∗
(
Cred(−;D)
)
,
Ktop
−∗
(
Γ,Cred(−;D)
)
and K−∗
(
Cred(−;D ⊗ K) ⋊ Γ
)
, respectively, in such a way that the cor-
responding sequences (5.3) are isomorphic to the long exact sequences induced by (6.1).
Proof. We reuse the proof of Lemma 4.20, but this time with the σ-Γ-C∗-algebras I = C0(X ;D)
⊗ K, A = Cred
0 (X ;D), B :=
∏
πMs(D) and J :=
⊕
π(D ⊗ K). Note that here we have
Cone
(
ι′ :
⊕
π
(D ⊗ K)→ C0(X ;D)
)
∼= C0(O+X ;D)⊗ K
and that the ∗-homomorphism Cone(ι′)→
⊕
π(D ⊗ K) = C0(π;D)⊗ K is exactly the evaluation
at the apices of the cones. The claim follows. ■
The Γ-equivariant coarse K-theory is the target of the so-called coassembly map, which we
recall next. Let X be a proper metric isometric Γ-space. Then for any discretization X ′ as
in Lemma 2.26, all the Rips complexes Pn(X
′) = PEn(X
′) carry an obvious Γ-isocoarse coarse
structure compatible with the topology such that all of the inclusionsX ⊃ X ′ ⊂ Pm(X ′) ⊂ Pn(X
′)
form ≤ n are coarse equivalences. It follows that cred(P(X ′);D) := lim←−n
cred(Pn(X
′);D) is a Γ-C∗-
algebra isomorphic to cred(X;D), because all of the maps in the inverse system are isomorphisms.
52 C. Wulff
Furthermore we define the σ-Γ-C∗-algebra cred(P(X ′);D) := lim←−n
cred(Pn(X
′);D). By [10] they
fit into a short exact sequence
0→ C0(P(X ′);D ⊗ K)→ cred(P(X ′);D)→ cred(P(X ′);D) ∼= cred(X;D)→ 0.
If A ⊂ X is a closed subspace and a suitable pair of discretizations (X ′, A′) has been chosen,
then we obtain analogously the short exact sequence
0→ C0(P(X ′) \ P(A′);D ⊗ K)→ c(P(X ′),P(A′);D)→ c(X,A;D)→ 0,
where c(P(X ′),P(A′);D) := ker(c(P(X ′);D)→ c(P(A′);D)).
Definition 6.5. The connecting homomorphisms in K-theory
µ∗ : K1−∗
(
cred(X;D)
)
→ KX∗(X;D),
K1−∗(c(X,A;D))→ KX∗ (X,A;D),
µ∗top : Ktop
1−∗
(
Γ; cred(X;D)
)
→ KX∗
Γ(X;D),
Ktop
1−∗(Γ; c(X,A;D))→ KX∗
Γ(X,A;D),
µ∗⋊ : K1−∗
(
cred(X;D)⋊ Γ
)
→ KX∗
Γ(X;D),
K1−∗(c(X,A;D)⋊ Γ)→ KX∗
Γ(X,A;D),
associated to the above short exact sequences are all called co-assembly maps.
The three co-assembly maps in the absolute case have originally been defined in [10, 11, 14],
respectively.
It is usually µ∗top which is of interest, because it is known to be an isomorphism in many
important cases, whereas not much can be said about µ∗⋊ [12, p. 72]. According to [14, Lemma 5.2],
the co-assembly map µ∗top factors through µ∗⋊ in the absolute case, but the same also holds true
for the corresponding relative versions. Therefore, if one is only interested in surjectivity (as was
the case in [14]), then µ∗⋊ is good enough.
Theorem 6.6. The transgression maps associated to the cohomology theories K∗(−, D), K∗
Γ(−, D)
and K−∗(C0(−;D ⊗ K)⋊ Γ) factor through the assembly maps as follows:
K̃∗−1(∂X;D)→ K1−∗
(
cred(X;D)
)
→ KX∗(X;D),
K∗−1(∂X, ∂A;D)→ K1−∗(c(X,A;D))→ KX∗(X,A;D),
K̃∗−1
Γ (∂X;D)→ Ktop
1−∗
(
Γ; cred(X;D)
)
→ KX∗
Γ(X;D),
K∗−1
Γ (∂X, ∂A;D)→ Ktop
1−∗(Γ; c(X,A;D))→ KX∗
Γ(X,A;D),
K1−∗
(
Cred(∂X;D)⋊ Γ
)
→ K1−∗
(
cred(X;D)⋊ Γ
)
→ KX∗
Γ(X;D),
K1−∗(C0(∂X \ ∂A;D ⊗ K)⋊ Γ)→ K1−∗(c(X,A;D)⋊ Γ)→ KX∗
Γ(X,A;D).
Thus, if a transgression map is an isomorphism (e.g., because P(X ′) is contractible or P(A′) is
a deformation retract of P(X ′)), then the corresponding co-assembly map is surjective.
Equivariant Coarse (Co-)Homology Theories 53
Proof. The claims follow directly from the commutative diagrams with exact rows
0 // C0(P(X ′);D ⊗ K) // Cred(P(X ′);D) //
��
Cred(∂X;D) //
��
0
0 // C0(P(X ′);D ⊗ K) // cred(P(X ′);D) // cred(X;D) // 0
0 // C0(P(X ′) \ P(A′);D) //
��
C0(P(X ′) \ P(A′);D) //
��
C0(∂X \ ∂A;D) //
��
0
0 // C0(P(X ′) \ P(A′);D ⊗ K) // c(P(X ′),P(A′);D) // c(X,A;D) // 0,
which exist because functions on a Higson-dominated compactification restrict to functions of
vanishing variation. ■
6.4 Coarse K-homology and assembly
Using Γ-equivariant E-theory,14 we define the Γ-equivariant K-homology with coefficients in
a separable Γ-C∗-algebra D as the functors
KΓ
∗ (−;D) := EΓ
∗ (C0(−);D).
Note that this definition requires the function algebras C0(−) to be separable, so we only consider
these functors on the full subcategory of LCH+
Γ of all second countable locally compact Γ-spaces.
In practice, this is only a minor restriction to the theory of Section 5, because we are mainly
interested in two special cases: First, Rips complexes of discretizations of countably generated
proper isocoarse Γ-spaces of bornologically bounded geometry are always second countable and,
second, their compactification with a Higson dominated corona is second countable, if the corona
is second countable, i.e., a compact metrizable space. The latter is an ubiquitous assumption in
applications.
Due to the well-known properties of bivariant K-theory (cf. [15, Chapter 6]), the functors
KΓ
∗ (−;D) are Γ-equivariant single-space homology theories on the above-mentioned category.
Furthermore, as direct limits preserve exact sequences, they can be extended to Γ-equivariant
single-space homology theories on the full subcategory of σLCH+
Γ consisting of all second countable
σ-locally compact Γ-spaces X =
⋃
n∈NXn by KΓ
∗ (X ;D) := limn→∞KΓ
∗ (Xn;D).
We want to compare the resulting transgression maps with the coarse assembly maps µ:
KXΓ
∗ (X;D) → K∗(C
∗
Γ(X;D)). The definition of the latter is based upon other pictures of
K-homology. We choose the one using Yu’s localization algebras [47] (see also [32] and [14]) for
our discussion. A lot about localization algebras has been proven in the comprehensive book [41],
but note that the “localized Roe algebras” considered there are slightly bigger than Yu’s original
localization algebras and hence a bit of care has to be taken when applying their results.
Following up our Section 4.4 about Roe algebras, let X be a proper metric space equipped
with a proper isometric Γ-action. We assume that sufficiently large Γ-X-module (H, ρ, u) has
been chosen and let H := H ⊗D be equipped with the induced representations. Recall that we
defined C∗
Γ(X;D) as the norm closure in B(H) of the ∗-subalgebra of all Γ-equivariant locally
compact operators of finite propagation, and if A is a subspace of X, then C∗
Γ(A ⊂ X;D) denoted
14Of course, KK-theory could also be used. In that case, a different description of the assembly maps would
need to be used and hence completely new proofs of our results would be required. However, KK-theory suffers
from similar technical shortcomings as E-theory and therefore no advantage over our approach is expected. In the
end, E-theory was primarily selected because of the author’s greater familiarity with this theory.
54 C. Wulff
the ideal generated by all operators which are supported in an R-neighborhood of A for some
R > 0 and C∗
Γ(X,A;D) := C∗
Γ(X;D)/C∗
Γ(A ⊂ X;D).
Definition 6.7. The localization algebra C∗
L,Γ(X;D) is the norm closure of the sub-∗-algebra of
Cb([1,∞),C∗
Γ(X;D)) of all bounded uniformly continuous functions L : [1,∞)→ C∗
Γ(X;D) such
that the propagation of L(t) is finite for all t ≥ 1 and converges to zero as t→∞. Furthermore, we
denote by C∗
L,Γ(A ⊂ X;D) ⊂ C∗
L,Γ(X;D) the ideal generated by all L for which there is a function
ε : [1,∞)→ (0,∞) with ε(t)
t→∞−−−→ 0 such that L(t) is supported in an ε(t)-neighborhood of A
for all t ≥ 1, and we define C∗
L,Γ(X,A;D) := C∗
L,Γ(X;D)/C∗
L,Γ(A ⊂ X;D).
An obvious variant of [41, Lemma 6.3.6], which can be proven with the same methods,
says that the inclusion A ⊂ X induces via the theory of covering isometries an isomorphism
K∗(C
∗
L,Γ(A;D)) ∼= K∗(C
∗
L,Γ(A ⊂ X;D)). Therefore we obtain a long exact sequence
· · · → K∗(C
∗
L,Γ(A;D))→ K∗(C
∗
L,Γ(X;D))→ K∗(C
∗
L,Γ(X,A;D))
→ K∗−1(C
∗
L,Γ(A;D))→ · · · . (6.2)
The relation between localization algebras and our definition of K-homology is the following.
As K(H) ∼= D ⊗ K(H), it is straightforward to verify (cf. [32, Corollary 4.2]) that
δ : C∗
L,Γ(X;D)⊗max C0(X)→ Cb([1,∞), D ⊗ K(H))
C0([1,∞), D ⊗ K(H))
,
L⊗ f 7→ [t 7→ L(t) ◦ (ρ(f)⊗ idD)]
defines an Γ-equivariant asymptotic morphism. It vanishes on the ideal C∗
L,Γ(A ⊂ X;D)⊗max
C0(X \A) and hence we also obtain an induced Γ-equivariant asymptotic morphism
δ : C∗
L,Γ(X,A;D)⊗max C0(X \A)→
Cb([1,∞), D ⊗ K(H))
C0([1,∞), D ⊗ K(H))
.
Both of them define canonical EΓ-theory elements in EΓ
0 (C
∗
L,Γ(X;D) ⊗max C0(X), D) and
EΓ
0 (C
∗
L,Γ(X,A;D)⊗max C0(X \A), D), respectively, which we shall also denote by the letter δ.15
We obtain the composed group homomorphism
∆: K∗(C
∗
L,Γ(X;D)) ∼= E∗(C,C∗
L,Γ(X;D))
−⊗idC0(X)−−−−−−−→ EΓ
∗ (C0(X),C∗
L,Γ(X;D)⊗max C0(X))
δ◦−−−→ EΓ
∗ (C0(X), D) = KΓ
∗ (X;D)
and similarily ∆: K∗(C
∗
L,Γ(X,A;D)) → KΓ
∗ (X,A;D) in the relative case. By exploiting the
homological properties of domains and targets of the maps ∆ (in particular that the domains
are functorial under uniformly continuous coarse Γ-maps via the theory of covering isometries,
that the maps ∆ are natural under such maps and that the connecting homomorphisms are in
both cases constructed via mapping cones) we see that the maps ∆ map (6.2) to the long exact
sequence of KΓ
∗ (−, D).
15Here we need EΓ-theory also for non-separable C∗-algebras. It can be defined in an ad hoc way as in [44,
Section 1], or even simpler by letting EΓ
∗ (A,B) for possibly non-separable A, B be obtained from the groups
EΓ
∗ (A
′, B′) by taking the direct limit over all separable B′ ⊂ B and the inverse limit over all separable A′ ⊂ A.
The relevant maps considered in the remainder of this section will be the same for both versions. Note that these
generalizations retain the composition and tensor products of EΓ
∗ -theory, but be aware that they are simply too
naive to preserve the long exact sequences.
Equivariant Coarse (Co-)Homology Theories 55
Theorem 6.8. The maps ∆ are isomorphisms in the absolute case for every proper metric
space X equipped with a proper isometric Γ-action and hence by a five lemma argument also in
the relative case.16
Proofs of some special cases of the theorem have been known for a long time. In the
non-equivariant case (Γ = 1) without coefficients (D = C), one proof is sketched in [41,
Construction 6.7.5, Exercises B.3.2 and B.3.3], albeit for a different version of the localization
algebra. Note, however, that this proof relies solely on homological properties and a comparison
on one-point spaces. Therefore, it goes through for original version of the localization algebras,
too, and even with arbitrary coefficient C∗-algebras D.
Another proof for the case Γ = 1 and D = C can be found in [32, Proposition 4.3] and relies
on Paschke duality. The author is indebted to a referee for pointing out a version of Paschke
duality for the equivariant KK-theory groups KKΓ
∗ (C0(X), D) which allows to prove the theorem
in full generality. Nevertheless, as localization algebras and E-theory are defined in a very similiar
manner, it would be interesting to know if the theorem can be proven directly without taking
the detour via Paschke duality.
Proof. It suffices to prove the theorem for separable D, because the general case follows by
taking direct limits over all separable sub-C∗-algebras. In this case, we can make use of the
Paschke duality for equivariant KK-theory that was discussed in [16, Appendix B].
An adjointable operator T on H is called pseudolocal if the commutators ρ(f)T − Tρ(f) are
compact for all f ∈ C0(X). Let D∗
Γ(X;D) denote the sub-C∗-algebra of B(H) generated by the
Γ-equivariant pseudolocal operators that have finite propagation and let D∗
L,Γ(X;D) denote the
sub-C∗-algebra of Cb([1,∞),D∗
Γ(X;D)) of all bounded uniformly continuous functions L : [1,∞)→
D∗(X;D) such that the propagation L(t) is finite for all t ≥ 1 and converges to zero as t→∞.
We are free to choose a very specific sufficiently large H = H ⊗D, namely H = HD := H ′ ⊗
ℓ2(N)⊗ ℓ2(Γ)⊗D for a non-degenerate Γ-X-module (H ′, ρ′, u′). It will be equipped with Γ-action
ε which acts diagonally on the first, third and fourth factor and the representation π of C0(X)
which acts on the first factor. For this choice, [16, Appendix B] contains the proof of isomorphisms
KKΓ
p (C0(X), D) ∼= Kp+1
(
D∗
Γ(X;D)
C∗
Γ(X;D)
)
∼= Kp+1
(
D∗
L,Γ(X;D)
C∗
L,Γ(X;D)
)
∼= Kp(C
∗
L,Γ(X;D))
for separable Γ-C∗-algebras D and proper metric spaces X equipped with a proper cocompact
isometric Γ-action.
The cocompactness assumption can be removed by performing slight modifications to the
aforementioned proof: Instead of Γ-propagation we only use metric propagation (cf. [16, items (ii)
and (iii) on p. 74]; in the co-compact case, finite Γ-propagation of a Γ-equivariant operator
is equivalent to finite metric propagation anyway). Furthermore, the proof utilises a cut-off
function c, i.e., a compactly supported continuous function X → [0, 1] such that
∑
γ∈Γ c(x·γ)2 = 1
for all x ∈ X. If the action is not co-compact, then we have to use the weaker condition on
the support of c that it intersect only finitely many Γ-translates of each compact subset. The
existence of such a cut-off function follows easily from the properness of the action. With these
modifications, the calculations in the proof of [16, Lemma B.7] go through. Just at the end of
that proof one has to perform another truncation a la [22, Sublemma 12.3.3] by an equivariant
partition of unity to turn the operator with finite Γ-propagation into one with finite metric
propagation. As [16, Lemmas B.14, B.15 and B.16] generalize to the non-cocompact case without
any problems, the abovementioned isomorphisms follow.
16Instead of Theorem 6.8, preprint versions of this article contained the “Assumption 6.8” that Γ and D have
been chosen for which the statement of this theorem holds. Thanks to a referee’s hint, we can now prove it in full
generality.
56 C. Wulff
Let us briefly recall the definition of the abovementioned isomorphisms in degree p = 0. Accord-
ing to the analogue of [16, Lemma B.7] in even degree17, every class in KKΓ
∗ (C0(X), D) can be rep-
resented by a cycle of the form
(
HD ⊕ HD,
(
0 U∗
U 0
)
, ϵ⊕ ϵ, π ⊕ π
)
with U ∈ D∗
Γ(X;D). Then U is a
unitary modulo C∗
Γ(X;D) and the Paschke duality isomorphism is defined by mapping this cycle to
the class represented by U . The second isomorphism then takes it to the class represented by a func-
tion t 7→ Ut which represents a unitary in D∗
L,Γ(X;D)/C∗
L,Γ(X;D) and such that U0 agrees with U
modulo C∗
Γ(X;D). Such a function can be constructed from the constant function U by applying
the procedure from the proof of [16, Lemma B.15]. The third isomorphism then maps it to the
class represented by the difference of the projection valued functions t 7→ Pt and t 7→ Q defined by
Pt :=
(
UtU
∗
t Ut(id−U∗
t Ut)
1
2
U∗
t (id−UtU
∗
t )
1
2 id−U∗
t Ut
)
and Q =
(
id 0
0 0
)
,
cf. [22, Proposition 4.8.10].
In order to prove the claim in even degrees, it now suffices to show that ∆ composed with these
isomorphisms is the canonical natural isomorphism KKΓ
0 (C0(X), D)
∼=−→ EΓ
0 (C0(X), D). The latter
is defined via [39, Theorem 2.2] and hence we have to verify two properties: That the composition
is natural in D and that it maps 1C0(X) ∈ KKΓ
0 (C0(X),C0(X)) to 1C0(X) ∈ EΓ
0 (C0(X),C0(X)).
Naturality is clear from the description of the three isomorphisms above and the definition of ∆.
For the verification of the other property, recall that 1C0(X) ∈ KKΓ
0 (C0(X),C0(X)) is repre-
sented by the cycle (C0(X)⊕0, 0, ε′⊕0, π′⊕0), where ε′ and π′ denote the canonical representations
of Γ and C0(X). After adding the degenerate cycle
(
HC0(X) ⊕ HC0(X),
(
0 id
id 0
)
, ϵ⊕ ϵ, π ⊕ π
)
, we
see that it is also represented by the cycle
(
HC0(X) ⊕ HC0(X),
(
0 U∗
U 0
)
, ϵ⊕ ϵ, π ⊕ π
)
for a projection
U : HC0(X)
∼= C0(X)⊕ HC0(X) → HC0(X) onto the second summand. Since U is a module homo-
morphism, it has propagation zero and U ∈ D∗
L,Γ(X; C0(X)). Furthermore we have UU∗ = id
and p := id−U∗U is a projection onto an isomorphically embedded copy of C0(X) in HC0(X).
For the calculation of the image of this class in EΓ
0 (C0(X), D) we can choose the constant
function t 7→ Ut := U . We end up with the element that is represented by the ∗-homomorphism
C0(X)→ K(HC0(X)), f 7→ π(f)p, and this is exactly 1C0(X).
The isomorphism in odd degrees now follows from the one in even degrees by replacing D
with the suspension D ⊗ C0(R). ■
The reason why we do not just choose the K-theory of the localization algebras as definition
of Γ-equivariant K-homology with coefficients is that we prefer to have a definition that works
also for non-proper group actions, but the proofs of the homological properties of K∗(C
∗
L,Γ(−;C))
in [41], in particular the existence of the all important covering isometries underlying functoriality,
rely heavily on properness. Note also that the equivariant Paschke duality that we used in the
proof of Theorem 6.8 works only for proper Γ-actions, so the two definitions might truely be
different in the non-proper case. Unfortunately, this makes the proof of Theorem 6.12 below
much more complicated than the argument using Paschke duality which was presented in [22,
Remark 12.3.8] for the non-equivariant case without coefficients.
Definition 6.9. Let (X,A) be a pair of proper metric spaces equipped with a proper isometric
Γ-action. The(uncoarsified) coarse assembly maps
µ : KΓ
∗ (X;D) ∼= K∗(C
∗
L,Γ(X;D))
(ev1)∗−−−−→ K∗(C
∗
Γ(X;D)),
µ : KΓ
∗ (X,A;D) ∼= K∗(C
∗
L,Γ(X,A;D))
(ev1)∗−−−−→ K∗(C
∗
Γ(X,A;D))
are induced by the evaluation at one.
17The cited lemma only considers the odd case, but the even case can be proven completely analogously.
Equivariant Coarse (Co-)Homology Theories 57
The coarsified coarse assembly map will be obtained by taking the direct limit over all
the uncoarsified assembly maps of the Rips complexes (Pn(X
′), Pn(A
′)) of a discretization
(X ′, A′) ⊂ (X,A). As we have chosen a description of the assembly maps which requires proper
metrics, we can’t content ourselves with the canonical topological coarse structures on the Rips
complexes, but we have to metrize them. Most authors equip the Rips complexes Pn(X
′) with
the spherical metric from [33], but they have the drawback that the inclusions X ′ ⊂ Pn(X
′)
are in general not coarse equivalences if the original space X was not a path-metric space.
Therefore, we use the Γ-invariant metrics from the following lemma, such that all of the inclusions
X ⊃ X ′ ⊂ Pm(X ′) ⊂ Pn(X
′) for m ≤ n are even isometric coarse equivalences. For a similar
approach to metrizing the Rips complexes, see [41, Section 7.2].
Lemma 6.10. We can assign to each discrete proper metric isometric Γ-space X ′ a Γ-invariant
metric dP(X′) on P(X ′) such that
• its restriction to P0(X
′) = X ′ is the original metric dX′,
• its restriction to each Pn(X
′) is a Γ-invariant proper metric inducing the already existing
topology on the complex,
• for all m ≤ n, Pm(X ′) is coarsely dense in Pn(X
′),
• if f : X ′ → Y ′ is a coarse Γ-map between discrete proper metric isometric Γ-spaces and X ′
is δ-separated for some δ > 0, i.e., the distance between each two distinct points of X ′ is at
least δ, then all of the restrictions
P(f)|Pn(X′) : Pn(X
′)→ P(Y ′)
are low-scale Lipschitz coarse maps. Here, by low-scale Lipschitz we mean that there are
constants L, δ′ > 0 such that the Lipschitz inequality
dP(Y ′)(f(x), f(y)) ≤ L · dP(X′)(x, y)
holds for all x, y ∈ Pn(X
′) with dP(X′)(x, y) < δ′.
The last property is very useful when working with the localization algebras, because the direct
description of the functoriality of their K-theory using families of covering isometries (in the
concrete version of [14, Definition 4.29 and Proposition 4.30]) is only available for uniformly
continuous coarse maps. Note that δ-separatedness is never a problem, because all discretizations
either already have this property by construction (cf. the proof of Lemma 2.6 applied to the
entourage Eδ) or can be thinned out such that they satisfy this property.
Proof. For R > 0, the R-spherical metric on a k-simplex is the path metric obtained by
identifying it with the subset
Sk
+(R) :=
{
(x0, . . . , xk) | x0, . . . , xk ≥ 0 ∧ x20 + · · ·+ x2k = R2
}
⊂ Rk+1.
On each Pn(X
′) with n ≥ 1 we first define the auxiliary metric d1Pn(X′) as the largest metric such
that its restriction to each simplex is at most the 2n-spherical metric. This metric is clearly
a proper path metric and it induces the given topology, but it may take the value ∞ and is not
yet the restriction of the metric dP(X′) that we are looking for. Also, note that X ′ is πn-dense
in Pn(X
′) with respect to d1Pn(X′). On P0(X
′) = X ′ we simply define d1P0(X′) = dX′ instead.
Thanks to the choice R = 2n, similar arguments to those in the proof of [33, Proposition 3.4(iv)]
(note also [34]) show that the inclusion X ′ ⊂ Pn(X
′) is distance non-decreasing with respect
to dX′ and d1Pn(X′) for each n ∈ N. Hence there is a largest metric d2Pn(X′) on Pn(X
′) which is
less or equal to d1Pn(X′) and restricts to dX′ on X ′. These new metrics have the property that
58 C. Wulff
the inclusions Pm(X ′) ⊂ Pn(X
′) are also distance non-decreasing with respect to d2Pm(X′) and
d2Pn(X′) for all m ≤ n.
Using all of these properties, it is now readily verified that there is a largest metric on dP(X′)
on P(X ′) such that all of the restrictions to Pn(X
′) are less or equal to d2Pm(X′) and this metric
clearly satisfies the first three properties on the list and is Γ-invariant.
Now, if f : X ′ → Y ′ and δ > 0 are as in the forth item on the list, then for each n ∈ N
there is an Sn ∈ N such that dX′(x, y) ≤ n implies dY ′(f(x), f(y)) ≤ Sn. Note that P(f)
restricts to a Sn
n -Lipschitz map from
(
Pn(X
′), d1Pn(X′)
)
to
(
PSn(Y
′), d1PSn (Y
′)
)
for every n ≥ 1
and, therefore, if x, y ∈ Pn(X
′) with dP(X′)(x, y) <
δ
2 , then dP(Y ′)(f(x), f(y)) ≤ L · dP(X′)(x, y)
with L := max
(
S1,
S2
2 , . . . ,
Sn
n
)
. ■
Definition 6.11. Taking direct limits over the uncoarsified assembly maps of the Rips complexes,
we obtain the (coarsified) coarse assembly maps,
µ : KXΓ
∗ (X;D) = lim−→
n∈N
KΓ
∗ (Pn(X
′);D)→ lim−→
n∈N
K∗(C
∗
Γ(Pn(X
′);D)) ∼= K∗(C
∗
Γ(X;D)),
µ : KXΓ
∗ (X,A;D) = lim−→
n∈N
KΓ
∗ (Pn(X
′), Pn(A
′);D)
→ lim−→
n∈N
K∗(C
∗
Γ(Pn(X
′), Pn(A
′);D)) ∼= K∗(C
∗
Γ(X,A;D)).
Theorem 6.12. Let X be a proper metric space X equipped with a proper isometric Γ-action
and let ∂X be a componentwise Higson dominated corona as defined in Definition 5.23. Then the
reduced transgression map factors through the assembly maps as follows:
KXΓ
∗ (X;D)
µ−→ K∗(C
∗
Γ(X;D))
ν−→ K̃Γ
∗−1(∂X;D).
Similarily, if A is a subspace of X and (∂X, ∂A) a pair of Higson-dominated coronas, then the
relative transgression map factors through the relative assembly maps:
KXΓ
∗ (X,A;D)
µ−→ K∗(C
∗
Γ(X,A;D))
ν−→ KΓ
∗−1(∂X, ∂A;D).
In particular, if the transgression maps are isomorphisms, then the assembly maps are injective.
Proof. We first prove the reduced version in detail. Write ∂X :=
∐
K∈Π0(X) ∂K and let
X :=
∐
K∈Π0(X)K be the decomposed locally compact Γ-space obtained by taking the Higson
dominated compactification K corresponding to ∂K of each coarse component K ∈ Π0(X). The
map ν is then defined as follows. The restrictions of functions in C0
(
X
)
to X have vanishing
variation on each coarse component and hence also on the whole space, so they commute with
operators in C∗
Γ(X;D) up to compact operators as in the proof of [35, Proposition 5.18]. Thus,
there is a Γ-equivariant ∗-homomorphism
C∗
Γ(X;D)⊗ C0(X)→ M(D ⊗ K(H))
D ⊗ K(H)
, T ⊗ f 7→ [T ◦ (ρ(f)⊗ idD)]
which vanishes on C∗
Γ(X;D) ⊗ C0(X) and hence factors through C∗
Γ(X;D) ⊗ C0(∂X). Note
that its restriction to C∗
Γ(X;D) ⊗ C0(Π0(X)) lifts to an Γ-equivariant ∗-homomorphism into
M(D ⊗ K(H)) when C0(Π0(X)) is considered as the sub-Γ-C∗-algebra of C0
(
X
)
via pull-back
under X → Π0(X). Identifying C0(O+(∂X)) with Cone(C0(Π0(X))→ C0(∂X)) and applying
the ∗-homomorphism above slice-wise, we obtain a Γ-equivariant ∗-homomorphism
δ′ : C∗
Γ(X;D)⊗ C0(O+(∂X))→ Cone
(
M(D ⊗ K(H))→ M(D ⊗ K(H))
D ⊗ K(H)
)
.
Equivariant Coarse (Co-)Homology Theories 59
The inclusion of D ⊗ K(H) as a canonical ideal in the mapping cone to the right induces an
isomorphism on EΓ-theory and hence we can define the map ν as the composition
K∗(C
∗
Γ(X;D)) ∼= E∗(C,C∗
Γ(X;D))
−⊗idC0(O+(∂X))−−−−−−−−−−→ EΓ
∗ (C0(O+(∂X)),C∗
Γ(X;D)⊗max C0(O+(∂X)))
δ′◦−−−−→ EΓ
∗
(
C0(O+(∂X)),Cone
(
M(D ⊗ K(H))→ M(D ⊗ K(H))
D ⊗ K(H)
))
∼= EΓ
∗−1(C0(O+(∂X)), D ⊗ K(H)) = K̃Γ
∗ (∂X;D).
The definition of ν is functorial under Γ-equivariant coarse maps and thus it identifies with the
corresponding maps νn associated to the locally compact Γ-spacesXn obtained fromXn := Pn(X
′)
by gluing ∂X to it. We have to show that the diagram
K∗(C
∗
L,Γ(Xn;D))
(ev1)∗ //
∼=∆
��
K∗(C
∗
Γ(Xn;D))
νn
��
KΓ
∗ (Xn;D)
∂ // K̃Γ
∗−1(∂X;D)
commutes and then the claim follows by taking the direct limit for n→∞. This will be done by
considering for each x ∈ K∗(C
∗
L,Γ(Xn;D)) the following diagram in the EΓ-category, in which all
solid arrows come from ∗-homomorphisms and all dashed arrows from asymptotic morphisms or
EΓ-theory elements:
C0(0, 1)
⊗C0(Xn)
≃ //
x⊗id
��
Cone
C0
(
O+
(
Xn
))
↓
C0(O+(∂X))
x⊗id
��
C0(0, 1)
⊗C0(O+(∂X))
oo
x⊗id
��
C∗
L,Γ(Xn;D)
⊗C0(0, 1)
⊗C0(Xn)
≃ //
δ⊗id
��
C∗
L,Γ(Xn;D)⊗ Cone
C0
(
O+
(
Xn
))
↓
C0(O+(∂X))
δ1
��
C∗
L,Γ(Xn;D)
⊗C0(0, 1)
⊗C0(O+(∂X))
oo
δ2⊗id��
C0(0, 1)⊗
D ⊗ K(H)
≃ // Cone
C0(0, 1]⊗M(D ⊗ K(H))
↓
Cone
(
M(D⊗K(H))→M(D⊗K(H))
D⊗K(H)
)
C0(0, 1)⊗
Cone
M(D⊗K(H))
↓
M(D⊗K(H))
D⊗K(H)
≃oo
C0(0, 1)⊗
D ⊗ K(H)
≃ // Cone
C0(0, 1]⊗ K(H)⊗D
↓
D ⊗ K(H)
≃
OO
C0(0, 1)⊗
D ⊗ K(H)
≃oo
≃
OO
The rows of the diagram are all of the form
J → Cone(A↠ A/J)← C0(0, 1)⊗A/J
for different C∗-algebras A with ideals J . The arrows going upwards are also induced by ∗-
homomorphisms of the form J → Cone(A ↠ A/J). All the arrows marked with “≃” are
60 C. Wulff
isomorphisms in EΓ-theory, because they are inclusions of ideals with contractible quotient
C∗-algebras. Furthermore, the mapping cone in the middle of the bottom row is itself isomorphic
to C0(0, 1)⊗D ⊗ K(H) and the maps leading into it then come from the inclusion of (0, 1) into
itself as the left or right half, up to orientation. Hence they represent the same element in
EΓ-theory up to a sign.
Using [41, Lemma 6.1.2], which clearly also holds true for operators on Hilbert modules, and
the fact that functions on Xn restrict to functions of vanishing variation on Xn, we see that
C∗
L,Γ(X;D)⊗max C
(
Xn
)
→ Cb([1,∞),M(D ⊗ K(H)))
C0([1,∞),M(D ⊗ K(H)))
,
L⊗ f 7→ [t 7→ L(t) ◦ (ρ(f)⊗ idD)]
is an equivariant asymptotic morphism. It clearly restricts to δ on the ideal C∗
L,Γ(X;D)⊗C0(Xn)
and also passes to an equivariant asymptotic morphism between the quotients
C∗
L,Γ(X;D)⊗ C(∂X)→ Cb([1,∞),M(D ⊗ K(H))/D ⊗ K(H))
C0([1,∞),M(D ⊗ K(H))/D ⊗ K(H))
.
Applying these constructions slicewise yields the asymptotic morphisms δ1 and δ2. In particular,
we note that δ2 is in fact given by the family of ∗-homomorphisms (δ′ ◦ evt)t∈[1,∞) and as such
it is 1-homotopic to the constant family (δ′ ◦ ev1)t∈[1,∞). Hence δ2 and δ′ represent the same
EΓ-theory class.
It is clear that the diagram commutes. The left column now is ∆(x) ∈ KΓ
∗+1((0, 1)×Xn;D) ∼=
KΓ
∗ (Xn;D) and the right column is νn ◦ (ev1)∗(x) ∈ KΓ
∗ (O+(∂X);D) ∼= K̃Γ
∗−1(∂X;D). The
middle column represents an element in KΓ
∗+1
(
O+
(
Xn
)
× {1} ∪ O(∂X) × (0, 1];D
)
which is
mapped to both ∆(x) and νn ◦ (ev1)∗(x) in the obvious way. By definition of the connecting
homomorphisms ∂ associated to the pair
(
O+
(
Xn
)
,O+(∂X)
)
via the mapping cone construction,
this shows exactly that ∂(∆(x)) = νn ◦ (ev1)∗(x).
The proof of the relative case works completely analogously but a lot simpler by showing that
for each x ∈ K∗(C
∗
L,Γ(Xn, An;D)) a diagram
C0(Xn \An)
≃ //
x⊗id
��
Cone
C0
(
Xn \An
)
↓
C0(∂X \ ∂A))
x⊗id
��
⊗C0(0, 1)
⊗C0(∂X \ ∂A)
oo
x⊗id
��
C∗
L,Γ(Xn, An;D)
⊗C0(Xn \An)
≃ //
δ
��
C∗
L,Γ(Xn, An;D)
⊗Cone
C0
(
Xn \An
)
↓
C0(∂X \ ∂A))
��
C∗
L,Γ(Xn, An;D)
⊗C0(0, 1)
⊗C0(∂X \ ∂A)
oo
��
D ⊗ K(H)
≃ // Cone
M(D ⊗ K(H))
↓
M(D⊗K(H))
D⊗K(H)
C0(0, 1)⊗ M(D⊗K(H))
D⊗K(H)
≃oo
commutes in the EΓ-category, where ∆(x) is the left column and ν((ev1)∗(x)) is the composition
of the right column with the bottom row. ■
Equivariant Coarse (Co-)Homology Theories 61
Acknowledgements
The author would like to thank Alexander Engel for insightful conversations. Furthermore, the
author is also very grateful to the anonymous referees for their meticulous inspection of the
manuscript, for their long list of comments and for pointing out numerous inaccuracies. Their
suggestions, critizism and partially also strong opinions had a significant positive influence on
this paper.
The research was supported by the DFG through the Priority Programme “Geometry at
Infinity” (SPP 2026; individual project “Duality and the coarse assembly map”, WU 869/1-1,
WU 869/1-2).
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1 Introduction
2 Coarse spaces
2.1 Coarse homotopies
2.2 Group actions on coarse spaces
3 Coarse (co-)homology theories
3.1 Flasqueness implies homotopy invariance
4 Examples
4.1 Ordinary coarse homology
4.2 Ordinary coarse cohomology
4.3 The stable Higson corona
4.4 K-theory of the Roe algebra
5 From topological to coarse (co-)homology theories
5.1 Categories of sigma-spaces
5.2 Generalized (co-)homology theories and the Rips complex
5.3 Single space (co-)homology theories and transgression maps
5.4 Reduced (co-)homology and reduced transgression
6 Further examples
6.1 Singular (co-)homology
6.2 Alexander–Spanier (co-)homology
6.3 Coarse K-theory and coassembly
6.4 Coarse K-homology and assembly
References
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| id | nasplib_isofts_kiev_ua-123456789-211730 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T18:18:05Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wulff, Christopher 2026-01-09T12:54:00Z 2022 Equivariant Coarse (Co-)Homology Theories. Christopher Wulff. SIGMA 18 (2022), 057, 62 pages 1815-0659 2020 Mathematics Subject Classification: 51F30; 5N35; 6L85 arXiv:2006.02053 https://nasplib.isofts.kiev.ua/handle/123456789/211730 https://doi.org/10.3842/SIGMA.2022.057 We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given, which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces. The author would like to thank Alexander Engel for insightful conversations. Furthermore, the author is also very grateful to the anonymous referees for their meticulous inspection of the manuscript, for their long list of comments, and for pointing out numerous inaccuracies. Their suggestions, critizism and partially also strong opinions had a significant positive influence on this paper. The research was supported by the DFG through the Priority Programme “Geometry at Infinity” (SPP 2026; individual project “Duality and the coarse assembly map”, WU 869/1-1, WU869/1-2). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Equivariant Coarse (Co-)Homology Theories Article published earlier |
| spellingShingle | Equivariant Coarse (Co-)Homology Theories Wulff, Christopher |
| title | Equivariant Coarse (Co-)Homology Theories |
| title_full | Equivariant Coarse (Co-)Homology Theories |
| title_fullStr | Equivariant Coarse (Co-)Homology Theories |
| title_full_unstemmed | Equivariant Coarse (Co-)Homology Theories |
| title_short | Equivariant Coarse (Co-)Homology Theories |
| title_sort | equivariant coarse (co-)homology theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211730 |
| work_keys_str_mv | AT wulffchristopher equivariantcoarsecohomologytheories |