Formal Aspects of Topological Complexity
We study the concept of topological complexity from the viewpoint of fibrewise Lusternik-Schnirelmann category and discuss certain formal aspects which include the equivalence of various descriptions, the axiomatic characterization, and the possibility to obtain a decomposition into $\Delta$-sets of...
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| author | Pavešić, P. Pavešić, P. |
| author_facet | Pavešić, P. Pavešić, P. |
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{
"author": "P. Pavešić",
"institution": "Faculty of Mathematics and Physics, University of Ljubljana"
}
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| description | We study the concept of topological complexity from the viewpoint of fibrewise Lusternik-Schnirelmann category and discuss certain formal aspects which include the equivalence of various descriptions, the axiomatic characterization, and the possibility to obtain a decomposition into $\Delta$-sets of different dimensions. |
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Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 56–66
Petar Pavešić
Formal Aspects of Topological
Complexity
We study the concept of topological complexity from the viewpoint of fibre-
wise Lusternik-Schnirelmann category and discuss certain formal aspects
which include the equivalence of various descriptions, the axiomatic char-
acterization, and the possibility to obtain a decomposition into ∆-sets of
different dimensions.
1The author was supported by the Slovenian Research Agency grant P1-02920101.
c© P. Petar Pavešić, 2013
Petar Pavešić 57
FORMAL ASPECTS OF TOPOLOGICAL COMPLEXITY
PETAR PAVEŠIĆ
Abstract. We study the concept of topological complexity from the view-
point of fibrewise Lusternik-Schnirelmann category and discuss certain formal
aspects which include the equivalence of various descriptions, the axiomatic
characterization, and the possibility to obtain a decomposition into ∆-sets of
different dimensions.
1. Introduction
The concept of topological complexity was introduced by M. Farber in [4, 5] in
his study of the navigation problem in robotics. Broadly speaking, the navigation
problem refers to the problem of finding a continuous motion that transforms a
mechanical system from some given initial position to a desired final position. To
give a mathematical formulation of this problem one introduces the so-called con-
figuration space, i.e. a topological space that describes all possible states of the
mechanical system. For such a configuration space X one then considers the space
XI of all continuous paths α : I → X, and the evaluation map ev : XI → X ×X
that to a path α assigns its end-points, ev(α) := (α(0), α(1)). A navigation plan
for X is a rule that takes as input a pair of points x, y ∈ X, and returns as out-
put a path α in X starting at x and ending at y. In other words, a navigation
plan is a section of the evaluation map, i.e. a function s : X ×X → XI such that
ev ◦ s = 1X×X . Observe that while the movement through the configuration space
is always assumed to be continuous with respect to the topology of the configura-
tion space, this is not necessarily the case for the navigation plan. In fact, one can
easily show that a continuous navigation plan exists if and only if X is contractible.
Thus, for non-contractible spaces one is naturally led to consider navigation plans
that are continuous only when restricted to subsets of X ×X.
Farber [4] exploited the fact that ev : XI → X ×X is a fibration, and defined the
topological complexity of path-connected space X to be the Schwarz genus [19] of
the fibration ev, i.e. the minimal n for which X×X can be covered by open subsets
U1, . . . , Un such that each of them admits a continuous section si : Ui → XI of ev.
A very similar approach was previously used by S. Smale [20] and A. Vassiliev [21]
in their investigation of the topological complexity of algorithms for finding roots
of polynomial equations. Observe that strictly speaking, the sections si : Ui → XI
do not determine a navigation plan for X because the elements of the open cover of
X must overlap, so over their intersections one has a multiple choice of navigation
1991 Mathematics Subject Classification. 55R70, 55M30.
Key words and phrases. topological complexity, fibrewise Lusternik-Schnirelmann category.
The author was supported by the Slovenian Research Agency grant P1-02920101.
1
58 Formal Aspects of Topological Complexity
plans. To avoid this difficulty, one may decompose X×X into disjoint subsets such
that the restriction of some global navigation plan to each of them is continuous.
Clearly for a non-contractible configuration space, every such global navigation plan
must be discontinuous, and that fact is sometimes described as the instability of the
navigation planning algorithm. Farber [5] tackled this problem and proved that the
topological complexity provides a suitable measure for the level of this instability.
It is clear from the definition that the topological complexity TC(X) is a homotopy
invariant of X, and so it has recently attracted a lot of interest among homo-
topy theorists. This resulted in a series of interesting developments, variations
and reformulations of the original idea. In particular, methods from the classical
Lusternik-Schnirellman (LS) category, in particular the Whitehead-Ganea approach
was developed in a series of papers [11], [12] and [13] by G. Calcines and L. Van-
dembroucq.
The alternative fibrewise LS category viewpoint was introduced by N. Iwase and M.
Sakai in [15], and further applied and developed in [8], [9] and [10]. The fibrewise
formulation avoids the use of function spaces, so the resulting theory has more
geometric flavour and opens the possibility of extensive application of the methods
of LS category to problems in topological complexity. In the first two section of
this paper we use the Iwase-Sakai approach to give a uniform overview of known
facts about the absolute and relative topological complexity together with slick and
efficient proofs. The remaining sections exploit the alternative approach to obtain
a couple of new results on the axiomatic approach to the topological complexity
and on some useful dimension-wise decompositions.
2. Topological complexity as fibrewise category
In this section we show that the topological complexity of X can be described in
terms of decompositions of the product X ×X into subsets that can be deformed
into the diagonal. and investigate the relations between different kinds of such
decompositions.
Let X be a path-connected space and let ev : XI → X × X be the evaluation
fibration ev(α) =
(
α(0), α(1)
)
. A subset F ⊆ X×X admits a continuous navigation
plan if there is a continuous map s : F → XI such that ev ◦ s = 1F . Various
descriptions of the topological complexity of X are related to different ways to
decompose of X×X into subsets that admit continuous navigation plans. We may
broadly distinguish four different approaches as follows.
1. Originally [4] the topological complexity of X was defined as the Schwarz genus
of the fibraton ev : XI → X×X. The Schwarz genus of a fibration p : E → B is the
minimal n for which B can be covered by n open sets U1, . . . , Un, such that each of
them admits a continuous local section si : Ui → E of p. The use of open covers is
standard in homotopy theory and allows direct comparison with other invariants.
For example, recall that cat(X), the Lusternik-Schnirelmann category of X, is the
minimal n for which X can be covered by n open sets U1, . . . , Un, such that each
Ui ↪→ X is null-homotopic, (i.e. each Ui can be deformed to a point inside X). One
then have the following basic estimate (cf. [7, Section 4.2])
(2.1) cat(X) ≤ TC(X) ≤ cat(X ×X) ≤ 2 cat(X)− 1 .
Petar Pavešić 59
2. For applications in robotics the unavoidable overlapping of the sets of an open
cover of X ×X sometimes creates problems because it introduces a level of ambi-
guity on which navigation plan should be used for pairs of points that lie in the
intersections. It is therefore often preferable to use partitions of X × X into dis-
joint subsets, so that the choice of the navigation plan is uniquely determined by
the input data. Furthermore, we want to avoid subspaces with bad local properties.
For that reason Farber [5] considered decompositions of X ×X as disjoint unions
of euclidean neighbourhood retracts. Recall that X is an euclidean neighbourhood
retract (ENR) if it is homeomorphic to a retract of an open subset of some eu-
clidean space Rn. More intrinsically, X is an ENR if it is locally compact, locally
contractible, and embeddable in some euclidean space (see [2, Section IV,8]). The
class of ENR’s contains all finite-dimensional cell complexes and all manifolds. Then
one can consider global navigation plans for X that are continuous when restricted
to the elements of some ENR-partition ofX×X (i.e. a decomposition into a disjoint
union of ENR’s). For example, Farber [5] proved that for a connected polyhedron
X the topological complexity of X equals the minimal n for which X ×X has an
ENR-partition into n subsets that admit continuous navigation plans.
3. Navigation plans that come up in applications are often defined locally, on
small subsets of the product X ×X. For example, we can describe simple-minded
navigation plans on a polyhedron X as follows. We first choose a maximal tree
T in the 1-skeleton of X. Then for each pair of vertices x, y ∈ X we define a
navigation plan on the product of open stars st(x)× st(y) by combining the unique
path in T between x and y with the straight segments in the respective stars. The
number of elements in such a cover of X × X by sets admitting navigation plans
is in general much bigger then TC(X). Since most of the elements are disjoint
one may aggregate them to produce covers with less elements but this is usually
impractical. There us however a different way to measure the complexity of such
navigation plans. Given a cover U of X the weight of U is the maximal number
of elements of U that have non-empty intersection. We will see later on that the
weights if such covers are bounded bellow by the topological complexity of X.
4. Finally we can combine locally defined navigation plans with the requirement
that their domains of definition are disjoint ENR’s. Given a global navigation plan
s : X ×X → XI and some cover {Fλ} of X ×X by mutually disjoint ENR’s, such
that the restrictions s|Fλ
are continuous, Farber [5] defined the order of instability
of this partition to be the weight of the cover {Fλ}. Once again, the topological
complexity turns out to be the precise lower bound for the orders of instability of
such partitions.
We now turn our attention from navigation plans to deformations of subsets of
X×X, starting from the following simple observation: every continuous navigation
plan s : F → XI by adjunction determines a homotopy ŝ : F × I → X ×X, given
by
ŝ(x, y, t) :=
(
x, s(x, y)(1− t)
)
.
Since s(x, y)(0) = x and s(x, y)(1) = y the homotopy ŝ is clearly a vertical (i.e along
the second factor) deformation of F to a subset of the diagonal ∆X = {(x, x) ∈
X ×X}. This was already noted in [6, Section 18] and further developed by Iwase
and Sakai in [15]. The main advantage of this alternative viewpoint is that a
deformation of a space is much easier to visualize than a map into a path space.
60 Formal Aspects of Topological Complexity
Every subset ofX×X that can be vertically deformed to a subset of the diagonal will
be called ∆-set. Various characterizations of topological complexity are summarized
in the following theorem.
Theorem 1. If X is an ENR then the topological complexity of X equals the
minimal n for which one (and hence all) of the following conditions is satisfied.
(1) There exists a cover of X ×X by n open ∆-sets.
(2) There exists a cover of X ×X by n closed ∆-sets.
(3) There exists an ENR-partition of X ×X into n disjoint ∆-sets.
(4) There exists a filtration ∅ = F0 ⊆ F1 ⊆ . . . ⊆ Fn = X×X by closed subsets,
such that each Fi − Fi−1 is a ∆-set.
(5) There exists a filtration ∅ = U0 ⊆ U1 ⊆ . . . ⊆ Un = X ×X by open subsets,
such that each Ui − Ui−1 is a ∆-set.
(6) There exists a filtration ∅ = C0 ⊆ C1 ⊆ . . . ⊆ Cn = X × X by locally
compact subsets, such that each Ci − Ci−1 is a ∆-set.
(7) X ×X admits a cover of weight n by open ∆-sets.
(8) X ×X admits a cover of weight n by closed ∆-sets.
(9) There exists an ENR-partition of X ×X into disjoint ∆-sets, whose order
of instability equals n.
Proof. (1) is just a reformulation of the definition of the Schwarz genus. (2) is
equivalent to (1) because as in the case of Lusternik-Schnirelmann category (cf. [17])
for spaces that are normal and neighbourhood retracts one can always work with
closed instead of open coverings, and vice versa. (3) follows from [7, Proposition
4.9]. (4)-(6) correspond to the characterizations of [7, Proposition 4.12]. (7),(8)
follow from [7, Corollary 4.14]. Finally (9) follows from [6, Theorem 13.1]. ¤
We are now going to relate the characterization (1) in the above theorem to a spe-
cial case of fibrewise Lusternik-Schnirelmann category. Take a ∆-set U ⊆ X ×X
and consider the projection π : X ×X → X of the product to the first factor. Re-
strictions of the homotopy that deforms U to the diagonal to the (possibly empty)
intersections Vx := U ∩pr−1(x) ⊆ {x}×X yields a family of homotopies indexed by
points of X that deform sets Vx within X to the point x. This precisely corresponds
to the idea of a fibrewise deformation of set to a point, on which the following defi-
nition of fibrewise Lusternik-Schnirelmann category is based (cf. [18]). A fibrewise
pointed space is a map p : E → B together with a section s : B → E: we view this
structure as a continuous family of pointed spaces p−1(b), each of them based at
the point s(b). Its fibrewise Lusternik-Schnirelmann category is the minimal n for
which E can be covered by open sets U1, . . . , Un such that for each i there is a
fibrewise homotopy deforming Ui to a subset of the section s(B) ⊂ E.
Let us consider the fibrewise pointed space over the base X whose total space is the
product X ×X, π : X ×X → X is projection to the first factor and the section is
given by the diagonal map ∆: X → X ×X. We will denote this fibrewise pointed
space by XnX where the semi-direct sign indicates that we have a ’twisted’ familly
of fibres indexed by the points of X, where the base ’acts’ on the fibres by sliding
the base-point. We may now conclude that TC(X) coincides with the fibrewise
Lusternik-Schnirelmann category of X nX.
Petar Pavešić 61
There are two important caveats regarding the role of the base-points (i.e. sections)
that one must keep in mind when discussing the fibrewise category as related to
the classical category. In the classical LS category the role of the base-points is
minor, because for spaces with nice local behaviour the pointed and unpointed
category coincide, and their value does not depend on the choice of the base-point.
In fact, one can use the homotopy extension property and arrange that all sets
of a categorical cover are deformed to the same point, and that all deformations
are stationary at that point. Contrary to that, two sections of a fibrewise space
may not be fibrewise homotopic, and the category with respect to one section can
be completely different from the category with respect to some other section. For
example the diagonal section of π : S2 × S2 → S2 is clearly not homotopic to the
constant section, and the fibrewise category of π : S2×S2 → S2 with respect to the
diagonal section equals the topological complexity TC(S2) = 3, while the fibrewise
category of π with respect to the constant section is the same as the ordinary
category cat(S2) = 2.
The second point is even more delicate. First of all, we define (following [18]) the
fibrewise pointed category of the fibrewise pointed space p : E → B with section
s : B → E as the minimal n for which E can be covered by open sets U1, . . . , Un
such that for each i s(B) ⊂ Ui and the fibrewise homotopy deforming Ui to s(B)
is stationary on s(B). The fibrewise pointed category is more adequate for the
application of the homotopy-theoretical methods (cf. [18, Section 6], [15]), but it
is not clear under what conditions the two notions coincide. In fact Iwase and
Sakai [15] proposed a proof that pointed fibrewise category equals the unpointed
fibrewise category for locally finite complexes but unfortunately their proof was
flawed, see the Errata [16]. At the moment the best result in this direction is by
A. Dranishnikov [3], who proved that the two versions of fibrewise category of X
coincide when certain assumptions on the dimension of X are satisfied.
3. Subspace complexity
In this section we consider the topological complexity of subspaces of X ×X. We
assume throughout that X is a Euclidean neighbourhood retract. Let A ⊆ X ×X
The subspace topological complexity of A, denoted TCX(A) is the least integer n for
which there exists a cover of A by n open ∆-subsets of X×X. Of course, instead of
covers by open sets we can use any of the equivalent descriptions of the topological
complexity listed in Theorem 1. It is easy to see that the subspace complexity
coincides with the relative complexity of A, which was defined in [7, Section 4.3] as
the Schwarz genus of the restriction over A of the evaluation fibration XI → X×X.
Let us list a few relations that follow immediately from the definition (most of them
already appeared in the literature, cf. [7], Chapter 4 and in particular Section 4.3).
First, we recover the topological complexity of X as
(3.1) TC(X) = TCX(X ×X).
If X ⊆ Y and A ⊆ B ⊆ X ×X then
(3.2) TCY (A) ≤ TCX(B).
62 Formal Aspects of Topological Complexity
If A,B ⊆ X ×X then
(3.3) TCX(A ∪B) ≤ TCX(A) + TCX(B).
Moreover, if A,B are separated open subsets of X ×X (i.e. A ∩ B = A ∩ B = ∅)
then
(3.4) TCX(A ∪B) = max{TCX(A),TCX(B)}.
The interplay between different characterizations given in Theorem 1 allows for
unified and efficient proofs of the various estimates for topological complexity. To
exemplify this approach we briefly summarize few most relevant results. we begin
with a lemma that gives us plenty of ∆-sets.
Lemma 2. Let X be a Euclidean neighbourhood retract.
(1) Any subspace of X ×X that can be deformed within X ×X into a ∆-set is
itself a ∆-set. In particular, every product of two categorical subsets of X
is a ∆-set (since it can be deformed to a point within X ×X).
(2) A union of a family of separated open ∆-sets is a ∆-set.
(3) If h : X nX → Y nY is a homeomorphism of fibrewise pointed spaces then
A is a ∆-set in X ×X if, and only if h(A) is a ∆-set in Y × Y .
Proof. (1) Let A ⊆ X × X, and let H : A × I → X × X be a deformation of A,
such that A′ = H1(A) is a ∆-set. If we denote by D′ : A′ × I → X ×X a vertical
deformation of A′ to the diagonal ∆X, than we obtain a vertical deformation D of
A to the diagonal by the formula
pr2D(x, y, t) :=
pr2 (H(x, y, 3t)) 0 ≤ t ≤ 1
3
pr2 (D
′(H(x, y, 1), 3t− 1)) 1
3 ≤ t ≤ 2
3
pr1 (H(x, y, 3− 3t)) 2
3 ≤ t ≤ 1
(2) Recall that a family of subsets of a topological space is separated if the closure of
each of them does not intersect the others. Clearly, when open ∆-sets are separated,
then their deformations to the diagonal combine to a continuous deformation of
their union to the diagonal.
3) A homeomorphism h : X×X → Y ×Y is a homeomorphism of fibrewise pointed
spaces if there is a homeomorphism h̄ : X → Y such that h̄ ◦ πX = πY ◦ h and
h ◦∆X = ∆Y ◦ h̄, so that the following diagram commutes
X ×X
h //
πX
²²
Y × Y
πY
²²
X
h̄
//
∆X
OO
Y
∆Y
OO
Then a deformationH : A×I → X×X of A to the diagonal ∆X yields a deformation
H : h(A)× I → Y × Y, H(y, y′, t) := h(H(h−1(y, y′), t)
of h(A) to the diagonal ∆Y . ¤
Petar Pavešić 63
Part (1) of the above Lemma implies that every categorical subset of X × X is
automatically a ∆-set, which immediately yields a relation between the subspace
topological complexity and subspace category:
(3.5) TCX(A) ≤ catX×X(A).
If B ⊆ X ×X can be deformed into some A ⊆ X ×X (i.e., there is a deformation
H : B × I → X ×X, such that H1(B) = H(B × 1) ⊆ A), then
(3.6) TCX(B) ≤ TCX(A).
In fact given a cover ofA by ∆-sets U1, . . . , Un, the pre-imagesH−1
1 (U1), . . . , H
−1
1 (Un)
cover B and are also ∆-sets by (1) of Lemma 2. As a special case, if B ⊆ X ×X
can be deformed to its subset A ⊆ B, then by 3.2
(3.7) TCX(A) = TCX(B).
Let X,Y be ENR’s with TC(X) = m and TC(Y ) = n. Then by Theorem 1 (5)
there exist a filtration ∅ = X0 ⊆ X1 ⊆ . . . ⊆ Xm = X ×X such that all Xi −Xi−1
are ∆-sets in X ×X and a filtration ∅ = Y0 ⊆ Y1 ⊆ . . . ⊆ Yn = Y × Y such that
all Yj − Yj−1 are ∆-sets in Y × Y . If we define Zk :=
⋃
i+j=k+1 Xi × Yj we obtain
a filtration ∅ = Z0 ⊆ Z1 ⊆ . . . ⊆ Zm+n−1 = (X × Y ) × (X × Y ). We directly
verify that Zk − Zk−1 =
∐
i+j=k+1(Xi −Xi−1)× (Yj − Yj−1) is a disjoint union of
separated ∆-sets, and conclude that
(3.8) TC(X × Y ) < TC(X) + TC(Y ).
Let G be a topological group. If U ⊆ G is an open categorical set, that can be
deformed to the unit e ∈ G then
⋃
g∈G{g} × gU is clearly a ∆-set in G × G. It
follows that a categorical cover of G gives rise to a cover of G×G by ∆-sets, hence
(3.9) TC(G) = cat(G).
4. Axiomatic characterization of topological complexity
Some of the properties listed in the previous section are sufficient to character-
ize precisely the subspace topological complexity among integer-valued functions
with similar properties. In fact, we are going to show that the formulas 3.2, 3.3
and 3.6, together with a normalization requirement are sufficient to determine the
topological complexity of a space. This approach is analogous to the axiomatic
characterization of the Lusternik-Schnirelmann category as in [1].
Let us define the abstract topological complexity on a space X to be a function
denoted tc(·) that assigns a positive integer to every non-empty subset A of X ×X
and satisfies the following properties:
(tc1) tc(∆X) = 1;
(tc2) If A ⊆ B ⊆ X ×X then tc(A) ≤ tc(B);
(tc3) If A,B ⊆ X ×X then tc(A ∪B) ≤ tc(A) + tc(B);
(tc4) If A,B ⊆ X × X, and B can be vertically deformed within X × X to a
subset of A
then tc(B) ≤ tc(A).
64 Formal Aspects of Topological Complexity
By the results from the previous section we know that the subspace topological
complexity TCX(·) satisfies the conditions for the abstract topological complexity.
We may now consider the set of all abstract topological complexities and order
them as follows: if tc1 and tc2 are two abstract topological complexities, let
tc1(·) ≤ tc2(·) ⇐⇒ tc1(A) ≤ tc2(A) for all A ⊆ X.
Let tc(·) be an abstract topological complexity, and let U be a non-empty ∆-subset
of X ×X. Then U can be vertically deformed to a subset of ∆X, so by (tc1) and
(tc4) we have tc(U) ≤ tc(∆X) = 1, therefore tc(U). Furthermore, If A ⊆ X ×X
can be covered by n open ∆-subsets U1, . . . , Un of X ×X then by (tc2) and (tc3)
tc(A) ≤ tc(U1 ∪ . . . ∪ Un) ≤ tc(U1) + . . .+ tc(Un) = n.
Since TCX(A) is precisely the minimal number of open ∆-subsets of X ×X that
are necessary to cover A we may conclude from the above discussion that
tc(A) ≤ TCX(A).
We have therefore proved the following result
Theorem 3. The subspace topological complexity TCX(·) is the maximal element
among all abstract topological complexities defined on subspaces of X ×X.
5. Dimension-wise ∆-sets
The standard minimal decompositions of Sn × Sn into a disjoint union of ENR
∆-sets that yield the topological complexities of the spheres are well known. For
odd-dimensional spheres we can take
A = {(x, y) ∈ Sn × Sn | x+ y 6= 0}
and
B = {(x, y) ∈ Sn × Sn | x+ y = 0},
and the dimensions are dim(A) = 2n and dim(B) = n. On the other side, for
even-dimensional spheres we may take
A = {(x, y) ∈ Sn × Sn | x+ y 6= 0},
B = {(x, y) ∈ Sn × Sn | x+ y = 0} − C,
and
C = {(N,−N), (−N,N)}
(where N ∈ Sn denotes the north pole), and the respective dimensions of the sets
involved are 2n, n and 0. One naturally wanders whether it is possible to achieve
the same (i.e. ∆-sets of different dimensions) in the general case. We are going to
prove this fact in the following form.
Theorem 4. Let X be a connected ENR and let A ⊆ X × X be an ENR subset
whose subspace topological complexity is TCX(A) = n. Then A can be decomposed
as a disjoint union A = X1 t . . . t Xn, where each Xi is an ENR ∆-set and
dim(A) = dim(X1) > dim(X2) > . . .dim(Xn) ≥ 0.
In particular, if X is a connected ENR whose topological complexity is TC(X) = n,
then X×X = X1t . . .tXn, where Xi are ENR ∆-sets and 2 dim(X) = dim(X1) >
dim(X2) > . . .dim(Xn) ≥ 0.
Petar Pavešić 65
The proof of the theorem is based on the following auxiliary result.
Lemma 5. For every ENR subset A ⊆ X × X there exists an ENR subset B ⊂
X ×X such that TCX(A) > TCX(B), dim(A) > dim(B) and (A−B) is a ∆-set.
Proof. For TCX(A) = 1 we take B := ∅.
Let TCX(A) = n and assume inductively that the claim holds for all B ⊆ X ×X
with TCX(B) < n. Let U1, . . . , Un be a cover of A by open ∆-sets in X. Then by
the normality of X, and by the properties of the small inductive dimension, we can
find an open set V1 in X such that
A− U2 − . . .− Un ⊆ V1 ⊆ V 1 ⊆ U1,
and satisfying the requirement dim(V 1−V1) < dim(A). We can furthermore find an
open cover V2, . . . , Vn of U2∪. . .∪Un such that V i ⊆ Ui and dim(V i−Vi) < dim(A).
Define B := (V 1 − V1) ∪ . . . ∪ (V n − Vn), so that clearly, dim(B) < dim(A).
Moreover, B is by the construction contained in the union U2 ∪ . . . ∪ Un, hence
TCX(B) < TCX(A). Each component of A−B is a ∆-set, as it contained in some
Ui. Since the components of A−B are separated 3.4 implies that A−B itself is a
∆-set, which concludes the proof. ¤
Proof. (of Theorem 4)
If TCX(A) = n we can inductively apply the above lemma to obtain spaces A =
A1 ⊃ A2 . . . ⊃ An ⊃ An+1 = ∅ such that dim(Ai) > dim(Ai+1) and (Ai − Ai+1)
are ENR ∆-sets. To obtain the decomposition stated in the theorem we let Xi :=
Ai −Ai+1. Moreover, it is clear that dim(A) = dimX1. ¤
If X is a polyhedron with TC(X) = n then the above argument can be easily
modified to obtain a filtration ∅ ≤ X1 ≤ . . . ≤ Xn = X ×X by polyhedra whose
dimension is strictly increasing, and such that each Xi − Xi−1 is a ∆-set. If X
is (p− 1)-connected then by Cellular approximation theorem every subcomplex of
dimension less then p is a ∆-set, which implies that dim(X2) ≥ p. It would be
interesting to know (at least for the case when p divides dim(X)) whether we can
extend further the analogy with the spheres and obtain a filtration of X × X as
above, by subpolyhedra whose dimensions are multiples of p.
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Faculty of Mathematics and Physics, University of Ljubljana
Jadranska 21
1000 Ljubljana, Slovenia
E-mail address: petar.pavesic@fmf.uni-lj.si
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| spelling | oai:trim.imath.kiev.ua:article-2752018-02-10T20:56:26Z Formal Aspects of Topological Complexity Формальні аспекти топологічної складності Pavešić, P. Pavešić, P. We study the concept of topological complexity from the viewpoint of fibrewise Lusternik-Schnirelmann category and discuss certain formal aspects which include the equivalence of various descriptions, the axiomatic characterization, and the possibility to obtain a decomposition into $\Delta$-sets of different dimensions. Ми вивчаємо поняття топологічної складності з точки зору пошарової категорії Люстерника-Шнірельмана та обговорюємо деякі формальні аспекти, що включають еквівалентність різних описів, аксіоматичну характеристику та можливість отримати розкладання в $ \Delta $ -множин різної розмірності. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/275 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 56-66 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 56-66 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 56-66 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/275/280 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Pavešić, P. Pavešić, P. Formal Aspects of Topological Complexity |
| title | Formal Aspects of Topological Complexity |
| title_alt | Формальні аспекти топологічної складності |
| title_full | Formal Aspects of Topological Complexity |
| title_fullStr | Formal Aspects of Topological Complexity |
| title_full_unstemmed | Formal Aspects of Topological Complexity |
| title_short | Formal Aspects of Topological Complexity |
| title_sort | formal aspects of topological complexity |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/275 |
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