A Decomposition of Twisted Equivariant -Theory
For a finite group, a normalized 2-cocycle α ∈ ²( , ¹) and a -space on which a normal subgroup acts trivially, we show that the α-twisted -equivariant -theory of decomposes as a direct sum of twisted equivariant -theories of parametrized by the orbits of an action of on the set of...
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| description | For a finite group, a normalized 2-cocycle α ∈ ²( , ¹) and a -space on which a normal subgroup acts trivially, we show that the α-twisted -equivariant -theory of decomposes as a direct sum of twisted equivariant -theories of parametrized by the orbits of an action of on the set of irreducible α-projective representations of . This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant K-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups ₂ₙ with ≥ 2, an even integer.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 041, 20 pages
A Decomposition of Twisted Equivariant K-Theory
José Manuel GÓMEZ and Johana RAMÍREZ
Escuela de Matemáticas, Universidad Nacional de Colombia, Medelĺın, Colombia
E-mail: jmgomez0@unal.edu.co, jramirezg@unal.edu.co
Received July 13, 2020, in final form April 15, 2021; Published online April 21, 2021
https://doi.org/10.3842/SIGMA.2021.041
Abstract. For G a finite group, a normalized 2-cocycle α ∈ Z2
(
G,S1
)
and X a G-space
on which a normal subgroup A acts trivially, we show that the α-twisted G-equivariant K-
theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized
by the orbits of an action of G on the set of irreducible α-projective representations of A.
This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math.
28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant K-theory. We also explore
some examples of this decomposition for the particular case of the dihedral groups D2n
with n ≥ 2 an even integer.
Key words: twisted equivariant K-theory; K-theory; finite groups
2020 Mathematics Subject Classification: 19L50; 19L47
1 Introduction
In the past few years there has been a growing interest in studying twisted K-theory motivated
by its appearance in string theory and also due to the celebrated theorem of Freed, Hokpins
and Teleman (see [5, Theorem 1]). In this article we study G-equivariant twisted K-theory
when G is a finite group. Our main goal is to show that, under suitable hypothesis, the canonical
decomposition theorem for projective representations can be used to obtain a decomposition
for twisted G-equivariant K-theory as a direct sum of other twisted equivariant K-theories,
thus generalizing the work in [6], where a similar decomposition was obtained for equivariant
K-theory.
Suppose that we have a short exact sequence of finite groups
1→ A→ G
π→ Q→ 1.
Let X be a compact and Hausdorff G-space on which A acts trivially and fix α a normalized
2-cocycle on G with values in S1. Associated to the cocycle α we have a central extension
1→ S1 → G̃α → G→ 1
and the action of G on X can be extended to an action of G̃α on X in such a way that the
central factor S1 acts trivially. The α-twisted G-equivariant K-theory of X, αK∗G(X), is con-
structed using G̃α-equivariant vector bundles on which the central factor S1 acts by multiplica-
tion of scalars. The main object of study in this work are the twisted K-groups αK∗G(X).
Representations of the group G̃α on which the central factor S1 acts by multiplication of sca-
lars are in a one to one correspondence with α-projective representations of G. Via this cor-
respondence we can use the classical tools of projective representations to study the twisted
K-groups αK∗G(X). To this end we show in Section 2 that, if Irrα(A) denotes the set of isomor-
phism classes of α-projective representations of A, then there is an action of G on Irrα(A) and
this action factors through an action of Q on Irrα(A). Given an isomorphism class [τ ] ∈ Irrα(A)
mailto:jmgomez0@unal.edu.co
mailto:jramirezg@unal.edu.co
https://doi.org/10.3842/SIGMA.2021.041
2 J.M. Gómez and J. Ramı́rez
let Q[τ ] denote the isotropy subgroup at [τ ]. Using Lemma 2.8 we show that we can associate
to each [τ ] ∈ Irrα(A) a normalized 2-cocycle defined on Q[τ ] with values in S1. This is precisely
the data needed to construct a twisted version of Q[τ ]-equivariant K-theory. With this in mind,
the main result of this article is the following theorem.
Theorem 1.1. Suppose that A is a normal subgroup of a finite group G. Let α be a normalized
2-cocycle on G with values in S1 and X a compact G-space on which A acts trivially. Then there
is a natural isomorphism
ΦX : αK∗G(X)→
⊕
[τ ]∈G\ Irrα(A)
βτ,αK∗Q[τ ]
(X),
[E] 7→
⊕
[τ ]∈G\ Irrα(A)
[
Hom
Ãα
(Vτ , E)
]
.
Explicit formulas for the cocycles βτ,α that appear in the previous theorem are described
in Section 2. Observe that when α is the trivial cocycle then αK∗G(X) agrees with K∗G(X) and
the previous decomposition agrees with [6, Theorem 3.2] in this case. Therefore Theorem 1.1
generalizes [6, Theorem 3.2]. In fact, Theorem 1.1 is proved using ideas similar to the ones used
to prove [6, Theorem 3.2].
We remark that the main idea used to prove Theorem 1.1 is well known and is often referred
to in the literature as “Mackey decomposition” or the “Mackey machine”. Moreover, similar
decomposition theorems have been obtained using these ideas for the case of equivariant K-
theory. To the knowledge of the authors the first instance where such a decomposition appears
is [10].1 In 1981 in his Ph.D. Thesis Wassermann used the “Mackey machine” in the context
of C∗-algebras to derive a decomposition for equivariant K-theory in the particular case that
a compact Lie group G acts on a locally compact space X with one orbit type. This means
that for every x ∈ X the isotropy subgroup Gx is conjugated to a fixed subgroup A (see [10,
Theorem 7]). The decomposition theorem obtained in [6] for equivariant K-theory is quite
similar to the one obtained by Wassermann with the difference that the result derived in [6]
works for a finite group G acting on a compact and Hausdorff space in X such a way that all
the isotropy subgroups Gx contain a fixed group A. In particular this means that the action
does not have to have just one orbit type. Recently, “Mackey decomposition” was used in [2]
to obtain a similar decomposition result for equivariant K-theory in the case of proper actions
of a compact Lie group G.
Theorem 1.1 could also be proved using the work of Freed, Hopkins and Teleman [5]. However,
we have chosen to prove it directly as to obtain an explicit description of this decomposition.
Also, we chose to work with finite groups to obtain explicit formulas for the cocycles used to twist
equivariant K-theory in this decomposition. Obtaining explicit formulas for such cocycles is one
of the main contributions of this work. Theorem 1.1 also holds in general for compact Lie groups,
a proof in this context can be obtained generalizing the work in [3].
The outline of this article is as follows. In Section 2 we review some basic definitions of projec-
tive representations that we use throughout this article. Section 3 is the main part of the article,
Theorem 1.1 is proved there. In Section 4 we use Theorem 1.1 to obtain a formula for the third
differential in the Atiyah–Hirzebruch spectral sequence for α-twisted G-equivariant K-theory
under suitable hypotheses. Finally, in Section 5 we explore some examples of Theorem 1.1 for
the particular case of the dihedral group D2n with n ≥ 1 an even integer.
1The authors would like to thank the editor for pointing out this reference that was not known previously to
them.
A Decomposition of Twisted Equivariant K-Theory 3
2 Projective representations
In this section we recall some basic definitions and properties of projective representations that
will be used throughout this article.
2.1 Basic definitions
Definition 2.1. Let G be a finite group and V a finite dimensional complex vector space.
A map ρ : G→ GL(V ) is called a projective representation of G if there exists a function
α : G×G→ C∗
such that
ρ(g)ρ(h) = α(g, h)ρ(gh) (2.1)
for all g, h ∈ G and ρ(1) = IdV .
Note that if ρ satisfies equation (2.1) then the function α defines a C∗-valued normalized
2-cocycle on G; that is, for all g, h, k ∈ G we have:
α(gh, k)α(g, h) = α(g, hk)α(h, k),
α(g, 1) = α(1, g) = 1.
To stress the dependence of ρ on V and α, we shall often refer to ρ as an α-representation of G
on the space V or, simply as an α-representation of G, if V is not pertinent to the discussion.
Remark 2.2. If α is the trivial cocycle; that is, if α(g, h) = 1 for all g, h ∈ G, then α-
representations of G are simply ordinary representations of G.
Definition 2.3. Suppose that ρi : G→ GL(Vi) with (i = 1, 2) are two α-representations.
1. A linear map ϕ : V1 → V2 is said to be a map of projective representations or a G-map
if for any g ∈ G and any v ∈ V1 we have ϕ(ρ1(g)v) = ρ2(g)ϕ(v). We write HomG(V1, V2)
for the set of G-morphisms from V1 to V2.
2. The α-representations V1 and V2 are said to be linearly equivalent or isomorphic if there
exists a G-map f : V1 → V2 that is an isomorphism of vector spaces. In other words,
V1 and V2 are isomorphic if there is a vector space isomorphism f : V1 → V2 such that
ρ2(g) = fρ1(g)f−1 for all g ∈ G.
Just as in the case of the ordinary theory of representations of groups we have similar notions
for projective representations such as irreducible representations and unitary representations.
Given a C∗-valued normalized 2-cocycle α on G we denote by Irrα(G) the set of isomorphism
classes of complex irreducible α-representations of G. If ρ : G → GL(V ) is an irreducible α-
representation of G then [ρ] ∈ Irrα(G) denotes the corresponding isomorphism class. Observe
that the direct sum of two α-representations is also an α-representation. Thus we can form the
monoid of isomorphism classes of α-representations. The α-twisted representation group of G,
denoted by Rα(G), is defined as the associated Grothendieck group. As an abelian group Rα(G)
is a free abelian group with one generator for each element in Irrα(G). We remark that the
classical results of representation theory such as complete reducibility and Schur’s lemma also
hold for the case of projective representations. We refer the reader to [7] for the basic theory
of projective representations.
4 J.M. Gómez and J. Ramı́rez
Example 2.4. Consider the dihedral group D2n of order 2n defined by
D2n =
〈
a, b | an = b2 = 1, bab = a−1
〉
.
For such groups we have
H2
(
D2n, S1
) ∼= {1 if n is odd,
Z/2 if n is even.
In this example we only consider the case where n is even as otherwise we will obtain usual
representations. Let n ≥ 2 be an even integer, ε a primitive n-th root of unity in C and let
α : D2n ×D2n → S1
be the function defined by
α
(
aj , akbl
)
= 1 and α
(
ajb, akbl
)
= εk for 0 ≤ j, k ≤ n− 1 and l = 0, 1.
The function α defines a normalized 2-cocycle on D2n with values in S1 whose corresponding
cohomology class is a generator in H2
(
D2n,S1
) ∼= Z/2. For each i ∈ {1, 2, . . . , n/2} put
Ai =
(
εi 0
0 ε1−i
)
and Bi =
(
0 1
1 0
)
.
Consider the map
τi : D2n → GL2(C)
defined by
τi
(
akbl
)
= AkiB
l
i for 0 ≤ k ≤ n− 1 and l = 0, 1.
These assignments determine the irreducible, non-equivalent α-representations of D2n so that
Irrα(D2n) =
{
[τ1], . . . , [τn/2]
}
and as an abelian group we have an isomorphism Rα(D2n) ∼=⊕n/2
i=1 Z[τi] (see for example [8, Chapter 5, Theorem 7.1]).
A key feature of projective representations is that we also have the following canonical decom-
position whose proof can be obtained in a similar way as in the case of regular representations.
Theorem 2.5 (canonical decomposition). Suppose that α is a normalized 2-cocycle of a finite
group G with values in S1. Let W be a finite-dimensional α-representation. Then the assignment
γ :
⊕
[ρ]∈Irrα(G)
Vρ ⊗HomG(Vρ,W )→W,
v ⊗ f 7→ f(v)
defines an isomorphism of α-representations.
Suppose now that α is a normalized 2-cocycle on G with values in S1. We can associate to α
a central extension of G by S1 in the following way. As a set define
G̃α =
{
(g, z) | g ∈ G, z ∈ S1
}
.
The product structure in G̃α is given by the assignment
(g1, z1)(g2, z2) := (g1g2, α(g1, g2)z1z2) .
A Decomposition of Twisted Equivariant K-Theory 5
This way G̃α is a compact Lie group that fits into a central extension
1→ S1 → G̃α → G→ 1.
Let ρ : G→ GL(V ) be an α-representation of G. If we define ρ̃ : G̃α → GL(V ) by ρ̃(g, z) = zρ(g)
then ρ̃ defines a representation of G̃α on which the central factor S1 acts by multiplication
of scalars. Conversely, if ρ̃ : G̃α → GL(V ) is a representation of G̃α on which the central fac-
tor S1 acts by multiplication of scalars then the function ρ : G→ GL(V ) given by ρ(g) = ρ̃(g, 1)
defines an α-representation of G. The above assignment defines a one to one correspondence
between α-representations of G and representations of G̃α on which the central factor S1 acts
by multiplication of scalars. Via this correspondence we will switch back and forth between
α-representations of G and representations of G̃α on which the central factor S1 acts by mul-
tiplication of scalars without explicitly mentioning it. In addition, if V1 and V2 are two α-
representations of G then having a map of projective representations f : V1 → V2 is equivalent
to having a linear map f : V1 → V2 that is G̃α-equivariant. With this correspondence in mind
we will also identify HomG(V1, V2) with Hom
G̃α
(V1, V2) without explicitly mentioning it.
2.2 Cocycles and projective representations
Suppose now that A is a normal subgroup of a finite group G so that we have a short exact
sequence
1→ A→ G
π→ Q→ 1.
Assume that α is a normalized 2-cocycle of G with values in S1, by restriction we can also see α
as a cocycle defined on A. Let us define an action of G on the set Irrα(A) in the following way.
Given ρ : A→ U(Vρ) an α-representation and g ∈ G we define g · ρ : A→ U(Vρ) so that if a ∈ A
we have:
g · ρ(a) = α(g−1a, g)α(g, g−1a)−1ρ(g−1ag) ∈ U(Vρ). (2.2)
Proposition 2.6. The above assignment defines a left action of G on Irrα(A). Furthermore,
for all b ∈ A, we have that b · ρ ∼= ρ so that the action of G on Irrα(A) factors to an action
of Q = G/A on Irrα(A).
Proof. First we show that g · ρ is an α-representation of A. Indeed, for all g ∈ G and a, b ∈ A
we have
(g · ρ(a))(g · ρ(b)) =
(
α
(
g−1a, g
)
α
(
g, g−1a
)−1
ρ
(
g−1ag
))(
α
(
g−1b, g
)
α
(
g, g−1b
)−1
ρ
(
g−1bg
))
= α
(
g−1a, g
)
α
(
g, g−1a
)−1
α
(
g−1b, g
)
α
(
g, g−1b
)−1
α
(
g−1ag, g−1bg
)
× ρ
(
g−1abg
)
= α
(
g, g−1a
)−1
α
(
g−1ab, g
)
α
(
g−1a, b
)
ρ
(
g−1abg
)
= α(a, b)α
(
g−1ab, g
)
α
(
g, g−1ab
)−1
ρ
(
g−1abg
)
= α(a, b)(g · ρ(ab)).
The above equalities are obtained using the 2-cocycle equation for α. Now, we show that this
definition satisfies the axioms of an action. If ρ : A → U(Vρ) is an α-representation, as α is
a normalized cocycle,
1 · ρ(a) = α(a, 1)α(1, a)−1ρ(a) = ρ(a).
Moreover, given g, h ∈ G and a ∈ A, we have
g · (h · ρ)(a) = α
(
g−1a, g
)
α
(
g, g−1a
)−1
(h · ρ)
(
g−1ag
)
= α
(
g−1a, g
)
α
(
g, g−1a
)−1
α
(
h−1g−1ag, h
)
α
(
h, h−1g−1ag
)−1
ρ
(
h−1g−1agh
)
6 J.M. Gómez and J. Ramı́rez
and
(gh) · ρ(a) = α
(
h−1g−1a, gh
)
α
(
gh, h−1g−1a
)−1
ρ
(
(gh)−1a(gh)
)
= α
(
h−1g−1a, gh
)
α
(
gh, h−1g−1a
)−1
ρ
(
h−1g−1agh
)
.
Manipulating the cocycle equation for α it can be proved that
α
(
g−1a, g
)
α
(
g, g−1a
)−1
α
(
h−1g−1ag, h
)
α
(
h, h−1g−1ag
)−1
= α
(
h−1g−1a, gh
)
α
(
gh, h−1g−1a
)−1
.
This implies that for all a ∈ A
g · (h · ρ)(a) = (gh) · ρ(a).
Finally, for a, b ∈ A expanding and using the cocycle equation we obtain
b · ρ(a) = ρ(b)−1ρ(a)ρ(b)
and thus b · ρ ∼= ρ as α-representations. �
Remark 2.7. The action of G on the set Irrα(A) given by equation (2.2) can be described
in an alternative way as follows. Suppose that α is a normalized 2-cocycle of G with values
in S1 and let G̃α be the central extension corresponding to the cocycle α. If ρ : A → U(Vρ)
is an α-representation of A then as explained before we have an associated representation
ρ̃ : Ãα → U(Vρ). Given (g, z) ∈ G̃α we obtain the Ãα-representation (g, z) · ρ̃ defined by
(g, z) · ρ̃(a,w) = ρ̃
(
(g, z)−1(a,w)(g, z)
)
.
This is a well-defined Ãα-representation such that the central factor S1 acts by multiplication
of scalars. Interpreting this Ãα-representation as an α-representation and using the cocycle
identities we obtain equation (2.2). Explicitly, for g ∈ G and a ∈ A we have
g · ρ(a) := (g, 1) · ρ̃(a, 1) = ρ̃
(
g−1ag, α
(
g−1a, g
)
α
(
g, g−1a
)−1)
= α
(
g−1a, g
)
α
(
g, g−1a
)−1
ρ
(
g−1ag
)
.
As above assume that A is a normal subgroup of a group G and Q = G/A. Fix an assignment
σ : Q→ G such that π(σ(q)) = q for all q ∈ Q with σ(1) = 1. We remark that the map σ is only
a set theoretical map so in particular it does not necessarily have to be a group homomorphism.
Suppose that ρ : A→ U(Vρ) is a complex irreducible α-representation with the property that
g · ρ is isomorphic to ρ for every g ∈ G (under the action defined in equation (2.2)). Under this
assumption, as σ(q) · ρ ∼= ρ we can find an element Mq ∈ U(Vρ) for each q ∈ Q such that
σ(q) · ρ(a) = M−1q ρ(a)Mq.
This means that
α
(
σ(q)−1a, σ(q)
)
α
(
σ(q), σ(q)−1a
)−1
ρ
(
σ(q)−1aσ(q)
)
= M−1q ρ(a)Mq
for all a ∈ A. We can choose M1 = 1 as σ(1) = 1 and σ(1)·ρ = ρ. Let Ãα be the central extension
of A by S1 associated to the cocycle α and ρ̃ : Ãα → U(Vρ) the corresponding representation.
Remember that ρ̃(a, z) = zρ(a) so we can define
σ(q) · ρ̃(a, z) := z(σ(q) · ρ(a)).
A Decomposition of Twisted Equivariant K-Theory 7
Therefore
σ(q) · ρ̃(a, z) = z
(
M−1q ρ(a)Mq
)
= M−1q zρ(a)Mq = M−1q ρ̃(a, z)Mq.
Define χ : Q×Q→ A by the equation
χ(q1, q2) = σ(q1q2)
−1σ(q1)σ(q2)
Note that χ(q1, q2) belongs to A since π(χ(q1, q2)) = 1 and the map χ is normalized in the sense
that χ(q1, q2) = 1 whenever q1 = 1 or q2 = 1. In addition, define τ : Q×Q→ S1 by
τ(q1, q2) = α
(
σ(q1q2)
−1, σ(q1)σ(q2)
)
α
(
σ(q1q2), σ(q1q2)
−1)−1α(σ(q1), σ(q2))
= α
(
σ(q1q2), χ(q1, q2)
)−1
α(σ(q1), σ(q2)).
Now, for q1, q2 in Q we notice that the element
ρ̃(χ(q1, q2), τ(q1, q2))M
−1
q2 M
−1
q1 Mq1q2 = τ(q1, q2)ρ(χ(q1, q2))M
−1
q2 M
−1
q1 Mq1q2
belongs to the center Z(U(Vρ)) ∼= S1. Define the map βρ,α : Q×Q→ S1 by the equation
βρ,α(q1, q2) := ρ̃(χ(q1, q2), τ(q1, q2))M
−1
q2 M
−1
q1 Mq1q2 . (2.3)
Lemma 2.8. If ρ : A→ U(Vρ) is a complex irreducible α-representation such that g · ρ ∼= ρ for
every g ∈ G then the map βρ,α : Q × Q → S1 defines a normalized 2-cocycle on Q with values
in S1.
Proof. If either q1 = 1 or q2 = 1 we have that τ(q1, q2) = 1 as α is normalized. Also, as χ is
normalized we have χ(q1, q2) = 1. Since we are choosing σ(1) = 1 and M1 = 1 it follows that
βρ,α(q1, q2) = 1 if either q1 = 1 or q2 = 1. Therefore either βρ,α is normalized. To finish we need
to prove that for every q1, q2, q3 ∈ Q we have
βρ,α(q1, q2q3)βρ,α(q2, q3) = βρ,α(q1q2, q3)βρ,α(q1, q2).
To see this note that as βρ,α(q1, q2) belongs to S1 we have that
Mq1q2 = βρ,α(q1, q2)Mq1Mq2 ρ̃(χ(q1, q2), τ(q1, q2))
−1.
Therefore, for q1, q2 and q3 in Q we have,
Mq1q2q3 = Mq1(q2q3) = βρ,α(q1, q2q3)Mq1Mq2q3 ρ̃(χ(q1, q2q3), τ(q1, q2q3))
−1
= βρ,α(q1, q2q3)Mq1βρ,α(q2, q3)Mq2Mq3 ρ̃(χ(q2, q3), τ(q2, q3))
−1
× ρ̃(χ(q1, q2q3), τ(q1, q2q3))
−1 = βρ,α(q1, q2q3)βρ,α(q2, q3)Mq1Mq2Mq3
×
[
ρ̃(χ(q1, q2q3), τ(q1, q2q3))ρ̃(χ(q2, q3), τ(q2, q3))
]−1
.
In a similar way, writing Mq1q2q3 = M(q1q2)q3 and expanding we obtain
Mq1q2q3 = βρ,α(q1q2, q3)βρ,α(q1, q2)Mq1Mq2Mq3
×
[
ρ̃(χ(q1q2, q3), τ(q1q2, q3))ρ̃(χ(q1, q2), τ(q1, q2))
]−1
.
This implies that
βρ,α(q1, q2q3)βρ,α(q2, q3) = βρ,α(q1q2, q3)βρ,α(q1, q2)
as we wanted to prove. �
8 J.M. Gómez and J. Ramı́rez
3 Decomposition of twisted equivariant K-theory
In this section we use the canonical decomposition of vector bundles to show that, under some
hypothesis, the α-twisted equivariant K-theory αK∗G(X) of a G-space X can be decomposed
as a direct sum of twisted equivariant K-theories parametrized by the orbits of the action of G
on Irrα(A) constructed on the previous section.
We start by recalling the definition of α-twisted equivariant K-theory that we will use. We fol-
low the treatment used in [1, Section 7.2]. Assume that G is a finite group acting on a compact
and Hausdorff space X. Let α be a normalized 2-cocycle on G with values in S1. Consider
1→ S1 → G̃α → G→ 1
the corresponding central extension. Observe that the action of G on X can be extended to
an action of G̃α in such a way that the central factor S1 acts trivially.
Definition 3.1. The α-twisted G-equivariant K-theory of X, denoted by αK0
G(X), is defined
as the Grothendieck group of the set of isomorphism classes of G̃α-equivariant vector bundles
over X on which S1 acts by multiplication of scalars on the fibers. For n > 0 the twisted groups
αKn
G(X) are defined as αK̃0
G(ΣnX+), where as usual X+ denotes the space X with an added
base point.
To obtain the desired decomposition of twisted equivariantK-groups we are going to construct
an equivalent formulation for such twisted K-groups following the work in [6]. For this suppose
that we have a short exact sequence of finite groups
1→ A→ G
π→ Q→ 1.
Assume that G acts on a compact and Hausdorff space X in such a way that A acts trivially.
Fix α ∈ Z2
(
G,S1
)
a normalized 2-cocycle. Notice that by restriction we can see any G̃α-
equivariant vector bundle over X as an Ãα-equivariant vector bundle, where Ãα denotes the
central extension associated to the cocycle α seen as a cocycle defined on A. Also, recall that
associated to an α-representation ρ : A→ U(Vρ) we have a representation ρ̃ : Ãα → U(Vρ).
Definition 3.2. Suppose that ρ : A → U(Vρ) is a complex irreducible α-representation.
A (G̃α, ρ)-equivariant vector bundle over X is a G̃α-vector bundle on which the central fac-
tor S1 acts by multiplication of scalars on E and the map
γ : Vρ ⊗Hom
Ãα
(Vρ, E)→ E,
v ⊗ f 7→ f(v)
is an isomorphism of Ãα-vector bundles on which S1 acts by multiplication of scalars.
In the above definition, if ρ is an α-representation of A then Vρ denotes the trivial Ãα-vector
bundle π1 : X × Vρ → X. Observe that if p : E → X is a G̃α-vector bundle on which the
central factor S1 acts by multiplication then, as we are assuming that A acts trivially on X,
it follows that for every x ∈ X the fiber Ex can be seen as a representation of Ãα on which
the central factor acts by scalar multiplication. Thus for every x ∈ X the fiber Ex can be seen
as an α-representation of A. With this point of view, a
(
G̃α, ρ
)
-equivariant vector bundle is
a G̃α-equivariant vector bundle p : E → X such that for every x ∈ X the fiber Ex is an α-
representation of A isomorphic to a direct sum of the α-representation ρ. Let Vect
G̃α,ρ
(X)
denote the set of isomorphism classes of
(
G̃α, ρ
)
-equivariant vector bundles, where two
(
G̃α, ρ
)
-
equivariant vector bundles are isomorphic if they are isomorphic as G̃α-vector bundles. Notice
that if E1 and E2 are two
(
G̃α, ρ
)
-equivariant vector bundles then so is E1 ⊕ E2. Therefore
Vect
G̃α,ρ
(X) is a semigroup. Following [6, Definition 2.2] we have the next definition.
A Decomposition of Twisted Equivariant K-Theory 9
Definition 3.3. Assume that G acts on a compact space X in such a way that A acts trivially
on X and let α be a normalized 2-cocycle α ∈ Z2
(
G,S1
)
. We define K0
G̃α,ρ
(X), the
(
G̃α, ρ
)
-
equivariant K-theory of X, as the Grothendieck construction applied to Vect
G̃α,ρ
(X). For n > 0
the group Kn
G̃α,ρ
(X) is defined as K̃0
G̃α,ρ
(ΣnX+).
As our next step we show that the previous definition can be described using the usual
definition of twisted equivariant K-theory provided in Definition 3.1. For this suppose that α is
a normalized 2-cocycle of G with values in S1. As above assume that A is a normal subgroup
of G and let Q = G/A. Let ρ be an irreducible α-representation such that g · ρ ∼= ρ for all
g ∈ G. Fix a set theoretical section σ : Q → G such that σ(1) = 1 as in the previous section.
We can extend σ to obtain a map σ̃ : Q → G̃α by defining σ̃(q) = (σ(q), 1) ∈ G̃α. Let βρ,α be
the 2-cocycle defined on Q with values in S1 constructed in equation (2.3). With this cocycle
we can consider the central extension
1→ S1 → Q̃βρ,α → Q→ 1.
With this in mind we have the following generalization of [6, Theorem 2.1].
Theorem 3.4. Suppose that ρ is an irreducible α-representation such that g ·ρ ∼= ρ for all g ∈ G.
Let X be a G-space such that A acts trivially on X. If p : E → X is a
(
G̃α, ρ
)
-equivariant vector
bundle, then Hom
Ãα
(Vρ, E) has the structure of a Q̃βρ,α-vector bundle on which the central
factor S1 acts by multiplication of scalars. Moreover, the assignment
[E]→ [Hom
Ãα
(Vρ, E)]
is a natural one to one correspondence between isomorphism classes of
(
G̃α, ρ
)
-equivariant vector
bundles over X and isomorphism classes of Q̃βρ,α-equivariant vector bundles over X for which
the central factor S1 acts by multiplication of scalars.
Proof. We are only going to provide a sketch of the proof as it follows the same steps used in
the proof of [6, Theorem 2.1].
Suppose that p : E → X is a G̃α-vector bundle. We give Hom
Ãα
(Vρ, E) an action of Q̃βρ,α .
Given f ∈ Hom
Ãα
(Vρ, E)x and q ∈ Q we define q • f ∈ Hom
Ãα
(Vρ, E)q·x by
(q • f)(v) = σ̃(q) · f
(
M−1q v
)
= (σ(q), 1) · f
(
M−1q v
)
,
where Mq ∈ U(Vρ) is the element chosen in the Section 2.2. It is easy to see that q • f is
Ãα-equivariant. Also, if q1, q2 ∈ Q then q1 • (q2 • f) is such that for v ∈ Vρ
q1 • (q2 • f)(v) = σ̃(q1)(q2 • f)
(
M−1q1 v
)
= σ̃(q1)σ̃(q2)f
(
M−1q2 M
−1
q1 v
)
= (σ(q1q2), 1)(χ(q1, q2), τ(q1, q2))f
(
M−1q2 M
−1
q1 v
)
= σ̃(q1q2)f(ρ̃
(
(χ(q1, q2), τ(q1, q2))M
−1
q2 M
−1
q1 v
)
= σ̃(q1q2)f
(
βρ,α(q1, q2)M
−1
q1q2v
)
= βρ,α(q1, q2)σ̃(q1q2)f
(
M−1q1q2v
)
= βρ,α(q1, q2)(q1q2 • f(v)).
We conclude that
q1 • (q2 • f)(v) = βρ,α(q1, q2)(q1q2 • f(v)).
The last equation allows us to define an action of Q̃βρ,α on Hom
Ãα
(Vρ, E) as follows. If (q, z) ∈
Qβρ,α and f ∈ Hom
Ãα
(Vρ, E)x define
(q, z) · f := z(q • f).
10 J.M. Gómez and J. Ramı́rez
Thus if v ∈ Vρ then
((q, z) · f)(v) = z
(
σ̃(q) · f
(
M−1q v
))
= σ̃(q) ·
(
zf
(
M−1q v
))
.
Unraveling the definitions, it is easy to see that this way Hom
Ãα
(Vρ, E) has the structure of
a Q̃βρ,α-equivariant vector bundle such that the central factor S1 acts by multiplication of scalars.
Suppose now that p : F → X is a Q̃βρ,α-equivariant vector bundle over X for which the
central S1 acts by multiplication of scalars. Given (g, z) ∈ G̃α, f ∈ Fx and v ∈ Vρ define
(g, z) · (v ⊗ f) := Mπ(g)ρ̃
(
σ̃(π(g))−1(g, z)
)
v ⊗ ((π(g), 1) · f) ∈ (Vρ ⊗ F )π(g)·x. (3.1)
The above assignment defines an action of G̃α on Vρ ⊗ F and Vρ ⊗ F becomes a G̃α-vector
bundle. Moreover for (a, z) ∈ Ãα, as M1 = 1, we have
(a, z) · (v ⊗ f) = M1ρ̃
(
σ̃(1)−1(a, z)
)
v ⊗ (1, 1) · f = ρ̃(a, z)v ⊗ f
so that Ãα acts on Vρ⊗F by the representation ρ̃; that is, p : Vρ⊗F → X is a
(
G̃α, ρ
)
-equivariant
vector bundle over X.
Finally, assume that p : E → X is a
(
G̃α, ρ
)
-equivariant vector bundle over X. By definition
the map
γ : Vρ ⊗Hom
Ãα
(Vρ, E)→ E,
(v, f) 7→ f(v)
is an isomorphism of Ãα-vector bundles. We may endow Vρ ⊗Hom
Ãα
(Vρ, E) with structure of
a G̃α-vector bundle. That the map γ is an isomorphism of vector bundles and its G̃α-equivariance
follows from next equations. For (g, z) ∈ G̃α we have
γ((g, z) · (v ⊗ f)) = γ
(
Mπ(g)ρ̃
(
σ̃(π(g))−1(g, z)
)
v
)
⊗ (π(g), 1) · f
= (π(g), 1) · f
(
Mπ(g)ρ̃
(
σ̃(π(g))−1(g, z)
)
v
)
= π(g) •f
(
Mπ(g)ρ̃
(
σ̃(π(g))−1(g, z)
)
v
)
= σ̃(π(g))· f
(
ρ̃
(
σ̃(π(g))−1(g, z)
)
v
)
= (g, z) · f(v) = (g, z) · γ(v ⊗ f).
The previous argument shows that γ : Vρ⊗Hom
Ãα
(Vρ, E)→ E is an isomorphism of G̃α-vector
bundles.
Now, if p : F → X is a Q̃βρ,α-equivariant vector bundle over X for which the central S1 acts
by multiplication of scalars, then by equation (3.1) we know that Vρ⊗F is a
(
G̃α, ρ
)
-equivariant
vector bundle. The canonical isomorphism of vector bundles
F → Hom
Ãα
(Vρ,Vρ ⊗ F ),
x 7→ fx : v 7→ v ⊗ x
is in fact Q̃βρ,α-equivariant, and therefore the vector bundles F and Hom
Ãα
(Vρ,Vρ ⊗ F ) are
isomorphic as Q̃βρ,α-equivariant vector bundles.
We conclude that the inverse map of the assignment [E] 7→ [Hom
Ãα
(Vρ, E)] is precisely the
map defined by the assignment [F ] 7→ [Vρ ⊗ F ]. �
As an immediate corollary of Theorem 3.4 we obtain the following identification of the (G̃α, ρ)-
equivariant K-theory groups of Definition 3.3 with the βρ,α-twisted Q-equivariant K-theory
groups provided in Definition 3.1.
A Decomposition of Twisted Equivariant K-Theory 11
Corollary 3.5. Let X be a G-space such that A acts trivially on X. Assume that ρ is an α-re-
presentation of A such that g · ρ ∼= ρ for every g ∈ G. Then the assignment
K∗
G̃α,ρ
(X)
∼=→ βρ,αK∗Q(X),
[E] 7→ [Hom
Ãα
(Vρ, E)]
defines a natural isomorphism.
Suppose now that α is a normalized 2-cocycle on G with values in S1. Consider the action
of G on Irrα(A) constructed in Section 2.2. Given [τ ] ∈ Irrα(A) let G[τ ] = {g ∈ G | g · τ ∼= τ}
denote the isotropy subgroup of the action of G at [τ ]. The group G[τ ] fits into the short exact
sequence
1→ A→ G[τ ]
π→ Q[τ ] → 1
and Q[τ ] = G[τ ]/A agrees with the isotropy of the group Q at [τ ]. Let
(
G̃[τ ]
)
α
be the central
extension corresponding to the cocycle α seen as a cocycle defined on G[τ ]. Therefore we have
the following commutative diagram of central extensions
1 −−−−→ S1 −−−−→
(
G̃[τ ]
)
α
−−−−→ G[τ ] −−−−→ 1
id
y y y
1 −−−−→ S1 −−−−→ G̃α −−−−→ G −−−−→ 1.
Assume that X is a compact and Hausdorff G-space on which A acts trivially. As before we can
extend the action of G on X to an action of G̃α on X in such a way that S1 acts trivially. Let
p : E → X be a G̃α-equivariant vector bundle on which S1 acts by scalar multiplication on the
fibers. As Ãα acts trivially on X each fiber of E can be seen as an α-representation of A. Using
fiberwise the canonical decomposition theorem for α-representations (Theorem 2.5) we see that
the assignment
γ :
⊕
[τ ]∈Irrα(A)
Vτ ⊗Hom
Ãα
(Vτ , E)→ E,
v ⊗ f 7→ f(v)
defines an isomorphism of Ãα-equivariant vector bundles. Using this decomposition we obtain
the following theorem.
Theorem 3.6. Under the above assumptions there is a natural isomorphism
ΨX : αK∗G(X)→
⊕
[τ ]∈G\ Irrα(A)
K∗
(G̃[τ ])α,τ
(X),
[E] 7→
⊕
[τ ]∈G\ Irrα(A)
[
Vτ ⊗Hom
Ãα
(Vτ , E)
]
.
This isomorphism is functorial on maps X → Y of G-spaces on which A acts trivially.
Proof. The proof of this theorem follows the same lines of the proof of [6, Theorem 3.1] so we
only provide an outline of the proof.
Let us show first that the map ΨX is well defined. To see this we have to show that if ρ
is an α-representation of A then Vρ ⊗ Hom
Ãα
(Vρ, E) has the structure of a
((
G̃[ρ]
)
α
, ρ
)
-vector
bundle. Following the notation of Section 2.2 fix a set theoretical section σ : Q[ρ] → G[ρ] for
12 J.M. Gómez and J. Ramı́rez
the projection map π : G[ρ] → Q[ρ] in such a way that σ(1) = 1. Also, for every q ∈ Q[ρ] fix
an element Mq ∈ U(Vρ) such that
σ(q) · ρ(a) = α
(
σ(q)−1a, σ(q)
)
α
(
σ(q), σ(q)−1a
)−1
ρ
(
σ(q)−1aσ(q)
)
= M−1q ρ(a)Mq.
This is possible as σ(q) ∈ G[ρ] so that we have σ(q) · ρ ∼= ρ. We can choose M1 = 1 as σ(1) = 1.
Now if (h, z) ∈
(
G̃[ρ]
)
α
and v ⊗ f ∈ Vρ ⊗Hom
Ãα
(Vρ, E) we define M(h,z) ∈ U(Vρ) by
M(h,z) := Mπ(h)ρ̃
(
σ̃(π(h))−1(h, z)
)
,
where ρ̃ and σ̃ are defined in a similar way as in Theorem 3.4. Observe that M(a,z) = ρ̃(a, z) for
all (a, z) ∈ Ãα. Moreover, given (h, z) ∈
(
G̃[ρ]
)
α
and v ⊗ f ∈ Vρ ⊗Hom
Ãα
(Vρ, E) we define
(h, z) ? (v ⊗ f) = M(h,z)v ⊗ (h, z) • f,
where (h, z) • f(w) = (h, z)f
(
M−1(h,z)w
)
. Unraveling the definitions it can be seen that this
defines an action of (G̃[ρ])α on Vρ⊗Hom
Ãα
(Vρ, E) in such a way that the central factor S1 acts
by multiplication of scalars. This way Vρ⊗Hom
Ãα
(Vρ, E) has the structure of a
(
G̃[ρ]
)
α
-vector
bundle and A acts by the α-representation ρ on the fibers so that
[
Vρ ⊗ Hom
Ãα
(Vρ, E)
]
∈
K∗
(G̃[ρ])α,ρ
(X). This shows that ΨX is well defined.
Next we show that ΨX is an isomorphism. For this write Irrα(A) = A1tA2t· · ·tAk, where
A1,A2, . . . ,Ak are the different G-orbits of the action of G on Irrα(A) defined in equation (2.2).
For every 1 ≤ i ≤ k define
EAi =
⊕
[τ ]∈Ai
Vτ ⊗Hom
Ãα
(Vτ , E).
Note that Vτ ⊗ Hom
Ãα
(Vτ , E) is an Ãα-equivariant vector bundle over X, so each EAi is also
an Ãα-equivariant vector bundle over X and the map
γ :
k⊕
i=1
EAi =
⊕
[τ ]∈Irrα(A)
Vτ ⊗Hom
Ãα
(Vτ , E)→ E
defines an isomorphism of Ãα-vector bundles. We are going to show that each EAi is a G̃α-vector
bundle and that the map γ is G̃α-equivariant. For this fix an index 1 ≤ i ≤ k and an irreducible
α-representation ρ : A → U(Vρ) such that [ρ] ∈ Ai. The elements in Ai can be written in the
form [g1 · ρ], . . . , [gri · ρ] for some elements g1 = 1, g2, . . . , gri ∈ G. Therefore
EAi =
ri⊕
j=1
Vgj ·ρ ⊗Hom
Ãα
(Vgj ·ρ, E).
We can give a structure of G̃α-space on EAi in the following way. Suppose that (g, s) ∈ G̃α
and that v ⊗ f ∈ Vρ ⊗Hom
Ãα
(Vρ, E)x. Decompose ggj in the form ggj = glh, where 1 ≤ l ≤ ri
and h ∈ G[ρ]. In other words gl is the representative chosen for the coset (ggj)G[ρ] and h =
g−1l ggj . Then
(g, s)(gj , 1) = (gl, 1)(h, z),
where z = sα(g, gj)α(gl, h)−1. We define
(g, s) ? (v ⊗ f) := M(h,z)v ⊗ (g, s) • f ∈
(
Vgl·ρ ⊗Hom
Ãα
(Vgl·ρ, E)
)
(g,s)·x,
A Decomposition of Twisted Equivariant K-Theory 13
where
(g, s) • f(w) = (g, s)f
(
M−1(h,z)w
)
= (g, s)(h, z)−1σ̃(π(h))f
(
M−1π(h)w
)
.
It can be seen that this defines an action of G̃α ∈ EAi for each 1 ≤ i ≤ k making the vector
bundle EAi into a G̃α-vector bundle in such a way that the central factor S1 acts by multiplication
of scalars. Furthermore, the map
γ :
k⊕
i=1
EAi → E
is an isomorphism of Ãα-vector and the map γ is G̃α-equivariant so that γ is an isomorphism
of G̃α-vector bundles. Now, the desired map ΨX can be seen as the direct sum ⊕ki=1Ψ
i
X choosing
for each Ai a representation [τi] ∈ Ai, where each Ψi
X is given by
Ψi
X : αK∗G(X)→ K∗
(G̃[τi])α,τi
(X),
[E] 7→
[
Vτi ⊗Hom
Ãα
(Vτi , E)
]
.
In what follows we will construct the map ζi : K
∗
(G̃[τi])α,τi
(X) → αK∗G(X) which will be the
right inverse of Ψi
X . Take ρ = τi and consider a vector bundle F ∈ Vect(G̃[ρ])α,ρ
(X). Fix
g1 = 1, g2, . . . , gr representatives for the different cosets in G/G[ρ]. Let
LF :=
r⊕
j=1
[
(gj , 1)−1
]∗
F.
Using ideas similar to the ones used in [6, Theorem 3.1] we can endow LF with the structure
of a G̃α-vector bundle on which the central factor S1 acts by multiplication of scalars in such
a way that the action of
(
G̃[ρ]
)
α
on
[
(g1, 1)−1
]∗
F ∼= F agrees with the given action of
(
G̃[ρ]
)
α
on F . We define
ζi : K∗
(G̃[τi])α,τi
(X)→ αK∗G(X),
[F ] 7→ [LF ].
Now, since we have the isomorphism Vρ⊗Hom
Ãα
(
Vρ,⊕rj=1
[
(gj , 1)−1
]∗
F
) ∼= F as
(
G̃[ρ]
)
α
vector
bundles. We obtain at the level of K-theory that ζi is a right inverse for Ψi so that ζ = ⊕ri=1ζi
is a right inverse for ΨX . In a similar way, using the work given above it can be seen that ζ is
also a left inverse so that the map ΨX is indeed an isomorphism.
To finish, we observe that functoriality follows from the fact that if τ is an α-representation
of A then the bundles Vτ ⊗ Hom
Ãα
(Vτ , f∗E) and f∗
(
Vτ ⊗ Hom
Ãα
(Vτ , E)
)
are canonically
isomorphic as ((G̃[τ ])α, τ)-equivariant bundles whenever f : Y → X is a G-equivariant map from
spaces on which A acts trivially. �
As a result of Theorem 3.6 and Corollary 3.5 we obtain the following theorem that is the
main result of this article.
Theorem 3.7. Suppose that A is a normal subgroup of a finite group G. Let α be a normalized
2-cocycle on G with values in S1 and X a compact G-space on which A acts trivially. Then there
is a natural isomorphism
ΦX : αK∗G(X)→
⊕
[τ ]∈G\ Irrα(A)
βτ,αK∗Q[τ ]
(X),
[E] 7→
⊕
[τ ]∈G\ Irrα(A)
[
Hom
Ãα
(Vτ , E)
]
.
14 J.M. Gómez and J. Ramı́rez
Here βτ,α is the 2-cocycle associated to τ and α as defined in equation (2.3). This isomorphism
is functorial on maps X → Y of G-spaces on which A acts trivially.
4 Atiyah–Hirzebruch spectral sequence
As an application of Theorem 3.7 we obtain a formula for the third differential in the Atiyah–
Hirzebruch spectral sequence for α-twisted G-equivariant K-theory under suitable hypotheses.
The treatment in this section generalizes the one given in [6, Section 5].
To start, assume that A is a normal subgroup of a finite group G and let Q = G/A. Suppose
that Q acts freely on a compact and Hausdorff space X. Let β : Q × Q → S1 be a normalized
2-cocycle with values in S1. As is explained in [6, equation 3.6], under these hypotheses the Q-
equivariant twisted K-group βK∗Q(X) can be described as a non-equivariant twisted K-group
over the space X/Q. To formulate this, let
1→ S1 → Q̃β → Q→ 1
be the central extension corresponding to β. Fix a separable Hilbert space H endowed with
a unitary linear action of Q̃β such that the central factor S1 acts by multiplication of scalars
and such that all the irreducible representations of this kind appear infinitely number of times
in H. Under these hypotheses, the space of Fredholm operators Fred(H) classifies β-twisted Q-
equivariant K-theory. Thus there is a natural isomorphism βK0
Q(X) ∼= [X,Fred(H)]Q. In ad-
dition, observe that the projective unitary group PU(H) is an Eilenberg–Maclane space of
type K(Z, 2). On the other hand, as S1 acts by multiplication of scalars on H, the action of Q̃β
on H induces a commutative diagram of central extensions of the form
1 −−−−→ S1 −−−−→ Q̃β −−−−→ Q −−−−→ 1
id
y φ̃β
y yφβ
1 −−−−→ S1 −−−−→ U(H) −−−−→ PU(H) −−−−→ 1.
Upon passage to classifying spaces, we obtain the map Bφβ : BQ → BPU(H) and BPU(H)
is an Eilenberg–Maclane space of type K(Z, 3) so that the homotopy class of Bφβ corresponds
to a cohomology class
[
βZ
]
∈ H3(BQ;Z). We remark that the class
[
βZ
]
agrees with the
class [β] ∈ H2(Q;S1) defined by the cocycle β under the standard identification H2
(
Q;S1
) ∼=
H3(BQ;Z). In addition, as X is a free Q-space there is a unique up to homotopy Q-equivariant
map X → EQ inducing a map h : X/Q → BQ at the level of the quotient spaces. This way
we obtain the continuous map fβ := Bφβ ◦ h : X/Q → BPU(H) which is precisely the data
needed to define the non-equivariant fβ-twisted K-groups. We remark that the cohomology
class associated to the homotopy class of the map fβ is precisely h∗
([
βZ
])
∈ H3(X/Q;Z).
By [6, equation (3.6)] we have an isomorphism
βK∗Q(X) ∼= K∗(X/Q; fβ). (4.1)
We refer the reader to [6, Section 3] for the details on this construction.
Assume now that X is a compact G-CW complex X in such a way that Gx = A for every
x ∈ A. Thus we are assuming that the G action on X has constant isotropy subgroups. This
is equivalent to asking that the group Q acts freely on X. Fix α ∈ Z2
(
G,S1
)
a normalized
2-cocycle. Let U = {Ui}i∈I (where I is a well ordered set) be a contractible slice cover of X
by G-invariant open sets. Thus for every sequence i1 ≤ · · · ≤ ip of elements in I with Ui1,...,ip :=
Ui1 ∩· · ·∩Uip nonempty we can find some element xi1,...,ip ∈ Ui1,...,ip such that the inclusion map
Gxi1,...,ip ↪→ Ui1,...,ip is a G-homotopy equivalence. It can be seen that such a contractible slice
A Decomposition of Twisted Equivariant K-Theory 15
cover exists for any compact G-CW complex. Using the G-cover U we can construct a spectral
sequence akin to the Atiyah–Hirzebruch spectral sequence that converges to αK∗G(X). This
spectral sequence can be constructed in the same way as in the case of equivariant K-theory
explained in [9, Section 5]. The E2-page of this spectral sequence is such that Ep,q2 = 0 if q
is odd. For even values of q we have that Ep,q2 = Hp
G(X,Rα(−)), the p-th Bredon cohomology
group Hp
G(X,Rα(−)) with coefficients in Rα(−). Here Rα(−) denotes the coefficient system
given by the α-twisted representation groups. Explicitly, this coefficient systems assigns Rα(H)
to the coset G/H. From here it follows automatically that the differential d2 is trivial and thus
E∗,∗2 = E∗,∗3 .
Theorem 4.1. Under the above hypotheses, the E3-term in the Atiyah–Hirzebruch spectral
sequence is such that
Ep,q3
∼=
{⊕
[τ ]∈Q\ Irrα(A)H
p
(
X/Q[τ ];Z
)
if q is even,
0 if q is odd.
Furthermore, the differential
d3 :
⊕
[τ ]∈Q\ Irrα(A)
Hp
(
X/Q[τ ];Z
)
→
⊕
[τ ]∈Q\ Irrα(A)
Hp
(
X/Q[τ ];Z
)
is defined coordinate-wise in such a way that for η ∈ Hp
(
X/Q[τ ];Z
)
we have
d3(η) = Sq3Zη − h∗
[
βZτ,α
]
∪ η.
Here βτ,α is the 2-cocycle associated to τ and α as defined in equation (2.3). Also, Sq3Z is the
composition of the maps β◦Sq2◦mod2, where mod2 is the reduction modulo 2, Sq2 is the Steenrod
operation, and β is the Bockstein map for the coefficient sequence 0→ Z 2−→ Z→ Z/2→ 0.
Proof. By assumption the action of G on X has constant isotropy, therefore the Bredon coho-
mology groups H∗G(X,Rα(−)) can be identified with the cohomology of the cochain complex
HomZ[G](C∗(X), Rα(A)). The group A acts trivially on both C∗(X) and Rα(A) so that this
cochain complex is isomorphic to HomZ[Q](C∗(X), Rα(A)). As a Q-module Rα(A) is a per-
mutation module. Therefore, as a Q-representation we have an isomorphism
Rα(A) ∼=
⊕
[τ ]∈Q\ Irrα(A)
Z
[
Q/Q[τ ]
]
.
Via this isomorphism we can identify Hp
G(X,Rα(−)) with
⊕
[τ ]∈Q\ Irrα(A)H
p
(
X/Q[τ ];Z
)
. For
even values of q it follows that Ep,q3 = Ep,q2 =
⊕
[τ ]∈Q\ Irrα(A)H
p(X/Q[τ ];Z). This proves the
first part of the theorem since we already know that for odd values of q we have Ep,q3 = Ep,q2 = 0.
To determine the third differential we use Theorem 3.7 to obtain an isomorphism
ΦX : αK∗G(X)→
⊕
[τ ]∈G\ Irrα(A)
βτ,αK∗Q[τ ]
(X).
By hypothesis, the group Q acts freely on X and thus for each [τ ] ∈ G\ Irrα(A) the group Q[τ ]
also acts freely on X. This together with (4.1) provides an isomorphism
αK∗G(X) ∼=
⊕
[τ ]∈Q\ Irrα(A)
K∗
(
X/Q[τ ]; fβτ,α
)
.
16 J.M. Gómez and J. Ramı́rez
For each [τ ] ∈ Q\ Irrα(A) passing to the quotient we obtain an open cover U/Q[τ ] :=
{
Ui/Q[τ ]
}
i∈I
of the quotient space X/Q[τ ]. Using the cover U/Q[τ ] we obtain a spectral sequence that con-
verges to K∗
(
X/Q[τ ]; fβτ,α
)
. The naturality of Theorem 3.7 implies that the spectral sequence
computing αK∗G(X) decomposes as a direct sum of the spectral sequences associated to the non-
equivariant twisted K-theories K∗
(
X/Q[τ ]; fβτ,α
)
. As it was pointed out above, the cohomology
class associated to the homotopy class of the map fβτ,α is precisely h∗
([
βZτ,α
])
∈ H3(X/Q;Z).
Using [4, Proposition 4.6] we conclude that the third differential in the Atiyah–Hirzebruch spec-
tral sequence to K∗
(
X/Q[ρ]; fβτ,α
)
is the operator
dρ3 : H∗(X/Q[τ ];Z)→ H∗+3
(
X/Q[τ ];Z
)
,
η 7→ dτ3(η) = Sq3Zη − h∗
[
βZτ,α
]
∪ η.
This proves the theorem. �
5 Examples
In this section we explore some examples of Theorems 3.6 and 3.7 for the dihedral groups D2n,
where n ≥ 2 an even integer.
We start by considering first the particular case where G = D8. The group D8 is generated
by the elements a, b subject to the relations a4 = b2 = 1 and bab = a3. Let α : D8×D8 → S1 be
the 2-cocycle defined by
α
(
al, ajbk
)
= 1 and α
(
alb, ajbk
)
= ij for 0 ≤ j, l ≤ 3 and k = 0, 1.
Note that α is a nontrivial normalized 2-cocycle such that its corresponding cohomology class
defines the generator of H2
(
D8; S1
) ∼= Z/2. By Example 2.4, taking n = 4, we know that up to
isomorphism D8 has two irreducible projective α-representations τ1 and τ2 defined by
τl
(
ajbk
)
= AjlB
k
l for 0 ≤ j, k ≤ 3 and l = 0, 1.
In the above definition we have
A1 =
(
i 0
0 1
)
, A2 =
(
−1 0
0 −i
)
and B1 = B2 =
(
0 1
1 0
)
.
With this in mind we are going to explore the following examples of Theorems 3.6 and 3.7.
Example 5.1. Suppose first that G = D8 and A = Z/4 = 〈a〉. Therefore
Q = G/A = {[1], [b]} ∼= Z/2.
Let us take X to be the space with only one point ∗ equipped with the trivial D8-action. In this
case αK∗D8
(∗) = Rα(D8), where Rα(D8) denotes the α-twisted representation group of D8.
As pointed out above τ1 and τ2 are the only irreducible α-representations of D8 and thus we
have an isomorphism of abelian groups
αK∗D8
(∗) ∼= Rα(D8) = Zτ1 ⊕ Zτ2.
On the other hand, observe that α|A is trivial so that
Irrα(A) = Irr(A) =
{
[1], [ρ],
[
ρ2
]
,
[
ρ3
]}
,
A Decomposition of Twisted Equivariant K-Theory 17
where ρ : A → C is the irreducible representation defined by ρ(a) = i. For the action of D8
on Irrα(A) we have
b · ρ(a) = α(ba, b)α(b, ba)−1ρ(bab) = α
(
a3b, b
)
α
(
b, a3b
)−1
ρ(a3) =
(
i3
)−1
i3 = 1
therefore b · ρ = 1. Moreover,
b · ρ2(a) = α(ba, b)α(b, ba)−1ρ2(bab) = α
(
a3b, b
)
α
(
b, a3b
)−1
ρ2
(
a3
)
=
(
i3
)−1(
ρ2(a)
)3
= −i
so that b · ρ2 = ρ3. We conclude that orbits of the D8 action on Irrα(A) are {[1], [ρ]} and{[
ρ2
]
,
[
ρ3
]}
. Thus we can choose [1] and
[
ρ2
]
as representatives for the elements in D8\ Irrα(A)
and G[1] = G[ρ2] = A. In this case Theorem 3.6 gives us an isomorphism
Ψ: Rα(D8) =α K∗D8
(∗)
∼=→ K∗
(G̃[1])α,1
(∗)⊕K∗
(G̃[ρ2])α,ρ
2(∗).
As G[1] = G[ρ2] = A and α restricted to A is trivial we have that
(
G̃[1]
)
α
=
(
G̃[ρ2]
)
α
= A × S1.
Therefore K∗
(G̃[1])α,1
(∗) ∼= Z1̃ and K∗
(G̃[ρ2])α,ρ
2
(∗) ∼= Zρ̃2. (Recall that ρ̃2 denotes the representa-
tion of
(
G̃[ρ2]
)
α
on which S1 acts by multiplication of scalars corresponding to ρ2 and similarly
for 1̃). For the representations τ1 and τ2 we have
τ1(a) =
(
i 0
0 1
)
and τ2(a) =
(
−1 0
0 −i
)
.
Thus as A-representations τ1 is isomorphic to 1⊕ ρ and τ2 is isomorphic to ρ2 ⊕ ρ3. Moreover,
in the isomorphism given by Theorem 3.6 we have
Ψ: Rα(D8) ∼= Zτ1 ⊕ Zτ2
∼=→ K∗
(G̃[1])α,1
(∗)⊕K∗
(G̃[ρ2])α,ρ
2
∼= Z1̃⊕ Zρ̃2,
τ1 7→ 1̃,
τ2 7→ ρ̃2.
On the other hand, by Theorem 3.7 we have an isomorphism
Φ: Rα(D8)
∼=→ β1,αK∗Q[1]
(∗)⊕ βρ2,αK∗Q[ρ2]
(∗).
As G[1] = G[ρ2] = A we have that Q[1] = Q[ρ2] = {1} is the trivial group. Therefore β1,α
and βρ2,α are the trivial cocycles and Theorem 3.7 gives us the isomorphism
Φ: Rα(D8) ∼= Zτ1 ⊕ Zτ2
∼=→ K∗{1}(∗)⊕K
∗
{1}(∗) ∼= Z⊕ Z.
Example 5.2. Suppose now that G = D8 and A = Z(D8) =
〈
a2
〉 ∼= Z/2. Therefore in this
case we have
Q = G/A = {[1], [b], [a], [ab]} ∼= Z/2⊕ Z/2.
As in the previous example α|A is trivial and
Irrα(A) = Irr(A) = {[1], [σ]},
where σ : A→ C is the representation defined by σ
(
a2
)
= −1. For the action of D8 on Irrα(A)
we have
b · σ
(
a2
)
= α
(
ba2, b
)
α
(
b, ba2
)−1
σ
(
ba2b
)
= α
(
a2b, b
)
α
(
b, a2b
)−1
σ
(
a2
)
= (−1)(−1) = 1
18 J.M. Gómez and J. Ramı́rez
therefore b · σ = 1. In particular the action of D8 on Irrα(A) is transitive and we can choose [1]
as a representative for the set D8\ Irrα(A). For the representation 1 we have G[1] = 〈a〉 ∼= Z/4.
If we take again X = ∗ then Theorem 3.6 gives us an isomorphism
Ψ: Rα(D8) ∼= Zτ1 ⊕ Zτ2
∼=→ K∗
(G̃[1])α,1
(∗).
As α is trivial on 〈a〉 we have that
(
G̃[1]
)
α
= 〈a〉 × S1. If ρ : 〈a〉 → C denotes the representation
defined by ρ(a) = i then 1̃ and ρ̃2 can be seen as
((
G̃[1]
)
α
, 1
)
-vector bundles over ∗ and
K∗
(G̃[1])α,1
(∗) ∼= Z1̃⊕ Zρ̃2.
Observe that
τ1(a
2) =
(
−1 0
0 1
)
and τ2(a
2) =
(
1 0
0 −1
)
.
Therefore as A-representations we have τ1 ∼= τ2 ∼= 1 ⊕ σ. However, as 〈a〉-representations τ1 is
isomorphic to 1⊕ ρ and τ2 is isomorphic to ρ2⊕ ρ3. It follows that in the isomorphism given by
Theorem 3.6 we have
Ψ: Rα(D8) ∼= Zτ1 ⊕ Zτ2
∼=→ K∗
(G̃[1])α,1
(∗) ∼= Z1̃⊕ Zρ̃2,
τ1 7→ 1̃,
τ2 7→ ρ̃2.
On the other hand, by Theorem 3.7 we have an isomorphism
Φ: Rα(D8)
∼=→ β1,αK∗Q[1]
(∗).
In this case Q[1] = 〈a〉/
〈
a2
〉 ∼= Z/2. The cocycle β1,α is the trivial cocycle and Theorem 3.7
gives us an isomorphism
Φ: Rα(D8) ∼= Zτ1 ⊕ Zτ2
∼=→ K∗Z/2(∗) = R(Z/2).
For the group Z/2 we have R(Z/2) ∼= Z1 ⊕ Zs, where s denotes the sign representation. The
isomorphism Φ maps τ1 to 1 and τ2 to s.
Example 5.3. Example 5.1 can easily be generalized to the dihedral groups D2n with n an even
number. Suppose then that n is an even number and let D2n be the group generated by the
elements a, b subject to the relations an = b2 = 1 and bab = a−1. Fix ε a primitive n-th root of
unity and let
α : D2n ×D2n → S1
be the function defined by
α
(
aj , akbl
)
= 1 and α
(
ajb, akbl
)
= εk for 0 ≤ j, k ≤ n− 1 and l = 0, 1.
The function α defines a normalized 2-cocycle on D2n with values in S1 whose correspond-
ing cohomology class is the generator in H2
(
D2n, S1
) ∼= Z/2. By Example 2.4 we know that
the irreducible projective α-representations of D2n are τi : D2n → GL2(C) for i = 1, . . . , n/2,
defined by
τi
(
akbl
)
= AkiB
l
i for 0 ≤ k ≤ n− 1 and l = 0, 1,
A Decomposition of Twisted Equivariant K-Theory 19
where
Ai =
(
εi 0
0 ε1−i
)
and Bi =
(
0 1
1 0
)
.
If we take X = ∗ endowed with the trivial D2n action we have
αK∗D2n
(∗) ∼= Rα(D2n) = Zτ1 ⊕ · · · ⊕ Zτn/2.
Take A = 〈a〉 ∼= Z/n so that Q = D2n/A = {[1], [b]} ∼= Z/2. Observe that α|A is trivial and thus
Irrα(A) = Irr(A) =
{
[1], [ρ],
[
ρ2
]
, . . . ,
[
ρn−1
]}
,
where ρ(a) = ε. For the action of D2n on Irrα(A) we have that
b · 1 = ρ, b · ρ2 = ρn−1, . . . , b · ρn/2 = ρn/2+1
so that orbits of the D2n action on Irrα(A) are
{
[1], [ρ]
}
,
{[
ρ2
]
,
[
ρn−1
]}
, . . . ,
{[
ρn/2
]
,
[
ρn/2+1
]}
.
We can choose [ρ],
[
ρ2
]
, . . . ,
[
ρn/2
]
as representatives for the set D2n\ Irrα(A) and we have G[ρ] =
G[ρ2] = · · · = G[ρn/2] = A. As α|A is trivial we have
(
G̃[ρi]
)
α
= A× S1 for i = 1, . . . , n/2. In this
case Theorem 3.6 gives us the isomorphism
Ψ: Rα(D2n) = Zτ1 ⊕ · · · ⊕ Zτn/2
∼=→
n/2⊕
i=1
K∗
(G̃[ρi])α,ρ
i(∗) ∼=
n/2⊕
i=1
Zρ̃i,
τi 7→ ρ̃i.
On the other hand, for i = 1, . . . , n/2 we have Q[ρi] = {1} and βρi,α is the trivial cocycle. In
this case Theorem 3.7 gives us an isomorphism
Φ: Rα(D2n) ∼= Zτ1 ⊕ · · · ⊕ Zτn/2
∼=→
n/2⊕
i=1
K∗{1}(∗) ∼= Zn/2.
Acknowledgements
The first author acknowledges and thanks the financial support provided by MINCIENCIAS
through grant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Cien-
cia, la Tecnoloǵıa y la Innovación. The second author acknowledges and thanks the financial
support provided by MINCIENCIAS through grant number 727 of the program Doctorados
nacionales 2015 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnoloǵıa y la Inno-
vación. Additionally, the authors would like to thank the referees and the editor for providing
useful comments that helped improve this manuscript.
References
[1] Adem A., Ruan Y., Twisted orbifold K-theory, Comm. Math. Phys. 237 (2003), 533–556,
arXiv:math.AT/0107168.
[2] Ángel A., Becerra E., Velásquez M., Proper actions and decompositions in equivariant K-theory,
arXiv:2003.09777.
[3] Ángel A., Gómez J.M., Uribe B., Equivariant complex bundles, fixed points and equivariant unitary bordism,
Algebr. Geom. Topol. 18 (2018), 4001–4035, arXiv:1710.00879.
[4] Atiyah M., Segal G., Twisted K-theory and cohomology, in Inspired by S.S. Chern, Nankai Tracts Math.,
Vol. 11, World Sci. Publ., Hackensack, NJ, 2006, 5–43, arXiv:math.KT/0510674.
https://doi.org/10.1007/s00220-003-0849-x
https://arxiv.org/abs/math.AT/0107168
https://arxiv.org/abs/2003.09777
https://doi.org/10.2140/agt.2018.18.4001
https://arxiv.org/abs/1710.00879
https://doi.org/10.1142/9789812772688_0002
https://arxiv.org/abs/math.KT/0510674
20 J.M. Gómez and J. Ramı́rez
[5] Freed D.S., Hopkins M.J., Teleman C., Loop groups and twisted K-theory I, J. Topol. 4 (2011), 737–798,
arXiv:0711.1906.
[6] Gómez J.M., Uribe B., A decomposition of equivariant K-theory in twisted equivariant K-theories, Inter-
nat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656.
[7] Karpilovsky G., Group representations, Vol. 2, North-Holland Mathematics Studies, Vol. 177, North-Holland
Publishing Co., Amsterdam, 1993.
[8] Karpilovsky G., Group representations, Vol. 3, North-Holland Mathematics Studies, Vol. 180, North-Holland
Publishing Co., Amsterdam, 1994.
[9] Segal G., Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.
[10] Wassermann A.J., Automorphic actions of compact groups on operator algebras, Ph.D. Thesis, University
of Pennsylvania, 1981.
https://doi.org/10.1112/jtopol/jtr019
https://arxiv.org/abs/0711.1906
https://doi.org/10.1142/S0129167X17500161
https://doi.org/10.1142/S0129167X17500161
https://arxiv.org/abs/1604.01656
https://doi.org/10.1007/BF02684591
1 Introduction
2 Projective representations
2.1 Basic definitions
2.2 Cocycles and projective representations
3 Decomposition of twisted equivariant K-theory
4 Atiyah–Hirzebruch spectral sequence
5 Examples
References
|
| id | nasplib_isofts_kiev_ua-123456789-211308 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T17:50:32Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gómez, José Manuel Ramírez, Johana 2025-12-29T11:08:10Z 2021 A Decomposition of Twisted Equivariant -Theory. José Manuel Gómez and Johana Ramírez. SIGMA 17 (2021), 041, 20 pages 1815-0659 2020 Mathematics Subject Classification: 19L50; 19L47 arXiv:2001.02164 https://nasplib.isofts.kiev.ua/handle/123456789/211308 https://doi.org/10.3842/SIGMA.2021.041 For a finite group, a normalized 2-cocycle α ∈ ²( , ¹) and a -space on which a normal subgroup acts trivially, we show that the α-twisted -equivariant -theory of decomposes as a direct sum of twisted equivariant -theories of parametrized by the orbits of an action of on the set of irreducible α-projective representations of . This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant K-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups ₂ₙ with ≥ 2, an even integer. The first author acknowledges and thanks the financial support provided by MINCIENCIAS through grant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación. The second author acknowledges and thanks the financial support provided by MINCIENCIAS through grant number 727 of the program Doctorados nacionales 2015 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación. Additionally, the authors would like to thank the referees and the editor for providing useful comments that helped improve this manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Decomposition of Twisted Equivariant -Theory Article published earlier |
| spellingShingle | A Decomposition of Twisted Equivariant -Theory Gómez, José Manuel Ramírez, Johana |
| title | A Decomposition of Twisted Equivariant -Theory |
| title_full | A Decomposition of Twisted Equivariant -Theory |
| title_fullStr | A Decomposition of Twisted Equivariant -Theory |
| title_full_unstemmed | A Decomposition of Twisted Equivariant -Theory |
| title_short | A Decomposition of Twisted Equivariant -Theory |
| title_sort | decomposition of twisted equivariant -theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211308 |
| work_keys_str_mv | AT gomezjosemanuel adecompositionoftwistedequivarianttheory AT ramirezjohana adecompositionoftwistedequivarianttheory AT gomezjosemanuel decompositionoftwistedequivarianttheory AT ramirezjohana decompositionoftwistedequivarianttheory |