Questions tagged [cw-complexes]

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Presentation complex of a finite perfect group and its features

Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions: Is there any special property of $X_G$ due to the group's perfectness? What can we say ...
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  • 851
1 vote
1 answer
132 views

Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?

Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the ...
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3 votes
1 answer
229 views

“Combinatorial” moves between cell complexes

EDITED: A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves. Is there a similar thing on finite cell complexes? That is, are ...
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  • 494
14 votes
2 answers
582 views

Is the decomposition of the homotopy type of a complex into a bouquet unique?

Is it true that if $A_1 ​​\vee A_2 \vee .. \vee A_n = B_1 \vee B_2 \vee .. \vee B_m$, where $A_i, B_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A_i$ and $...
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7 votes
2 answers
823 views

Does there exist a complete algebraic invariant of the homotopy type of a finite CW-complex?

Let $\mathrm{Cell}$ be the homotopy category of finite cell complexes. The main motive of my question Is it true that for any algebraic category $A$ there is no fully faithful functor $F: \mathrm{...
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2 votes
1 answer
83 views

Double coset decomposition for compact Lie groups

The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows \begin{...
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  • 645
3 votes
1 answer
157 views

Ehresmann's fibration theorem for CW or simplicial complexes

Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ? Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is ...
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3 votes
0 answers
181 views

CW-complexes that cannot be homotopically compressed

Definition. A CW-complex $A$ can be elementary compressed to a CW-complex $B$ if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according ...
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3 votes
0 answers
70 views

Description of Anderson-Putnam CW-complex construction

I have been trying to read the paper, Topological invariants fo substitution tiling and their associated $C^*$-algebras, to learn more about a construction of Anderson-Putnam complexes. However, it ...
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3 votes
0 answers
214 views

CW structure on $\mathrm{PU}(3)$/Heisenberg group

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where $H$ is the Heisenberg group of order 27 $G_{81}$ is the No. 9 group of order 81 (...
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111 views

Cohomology spectral sequence of a CW complex filtered by its skeletons

Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$ is a filtration of $X$ by its skeletons $X^i$. Now ...
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  • 159
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0 answers
140 views

When is the space of maps between varieties a finite CW complex?

$\DeclareMathOperator\Cont{Cont}$Given two algebraic varieties over $\mathbb{C}$ denoted by $X$ and $Y$ where $Y$ is projective and $X$ is either projective or affine/Stein. The space of continuous ...
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  • 4,926
11 votes
2 answers
626 views

Can we embed a closed manifold into a homotopy equivalent CW complex?

Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...
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  • 663
1 vote
3 answers
604 views

How can I construct a closed manifold from a finite CW complex?

If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...
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  • 851
6 votes
3 answers
711 views

Finite CW complex with finite non-abelian fundamental group and higher homologies zero

I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$ From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=...
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  • 851
2 votes
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135 views

Dimension range for non-torsion homotopy groups

Is there a constant $c$ for which the following is true? Let $X$ be a connected finite CW complex of dimension $d$. For any $i>cd$, the homotopy group $\pi_i(X)$ is torsion. What if we replace ...
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  • 2,032
3 votes
1 answer
168 views

Embedding CW-complexes into infinite-dimensional topological vector spaces

Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
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8 votes
1 answer
386 views

Whitehead product and a homotopy group of a wedge sum

Note : this is a crosspost from the Mathematics StackExchange, as suggested by this meta post. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-...
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1 vote
0 answers
82 views

Codimension one submanifold gives cofibrant pair

Let $M$ be a smooth manifold, and $N$ be an embedded smooth submanifold of $M$ with $\partial M=\varnothing=\partial N$. Suppose, $\dim M-\dim N=1$, and $N$ is a closed subset of $M$. Does the ...
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  • 265
3 votes
0 answers
118 views

Null-homotopic cellular loops are elementary null-homotopic?

I've got a 2-dimensional cell complex $X$ and a cellular closed loop $l \subset X$ that I happen to know is null-homotopic in $X$. There are some very simple sorts of homotopies of cellular loops (or ...
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  • 5,201
4 votes
1 answer
287 views

Cellular homology of the universal cover

Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$. Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...
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  • 893
13 votes
1 answer
629 views

Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?

The question is inspired by an answer to The concept of Duality It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
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12 votes
0 answers
193 views

Smooth dual cell structure

Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
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12 votes
0 answers
390 views

Covering image of a connected CW-complex need not be a CW-complex

This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
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  • 582
5 votes
2 answers
305 views

Regular or h-regular CW-complex structure for the Poincaré homology sphere

I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré ...
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  • 869
6 votes
2 answers
281 views

CW-presentation of configurations of points in plane and space

I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the ...
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  • 32.2k
1 vote
1 answer
128 views

can we take skeletons of covering maps to give new covering maps?

Let $X$ be an $n$-dimensional cell complex. We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$. Take the universal cover (or a general covering space) $\tilde X'$ of $X'$. ...
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  • 1,940
0 votes
1 answer
145 views

How do I show that any finite-dimensional (absolute) CW-complex $X$ is locally contractible?

I know that it holds even if $X$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.
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8 votes
1 answer
292 views

K-theory on finite-dimensional (possibly not finite) CW complexes

I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
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7 votes
1 answer
219 views

The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
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4 votes
0 answers
169 views

What does go wrong in Cellular homology if one considers projective limits of celullar complexes instead of CW-complexes?

Consider a nice topological space $X$ (e.g. the 3-sphere) and consider inside a decreasing sequence of compact subsets $(K_n)_{n\in\mathbb N}$ such that $K_\infty:=\bigcap_{n\in \mathbb N} K_n$ is 0-...
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2 votes
0 answers
283 views

What topological spaces can be realized as cell complexes?

What are the topological spaces can be realized as cell complexes, up to homeomorphism? It seems for instance that all manifolds can be built from cell complexes. It is clear however that one can ...
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  • 494
2 votes
0 answers
167 views

regular CW complex and incidence matrices

Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case. If it's ...
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  • 494
9 votes
0 answers
322 views

Homotopical characterization of CW complexes

Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$. Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
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  • 117
5 votes
1 answer
363 views

loop space of a finite CW-complex

Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...
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  • 893
6 votes
1 answer
149 views

Invariant neighborhood in a G-CW complex

Let $G$ be a discrete group and $X$ be a $G$-CW complex. For any $x\in X$ and open neighborhood $U$ of $x$, I am interested in the question that whether we can find a $G_x$-invariant open ...
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  • 391
9 votes
1 answer
301 views

Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?

Let $M$ be a connected open topological $d$-manifold (without boundary). Whitehead showed that if $M$ has a PL structure, there exists a subcomplex of dimension $\leq d-1$ onto which $M$ deformation ...
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  • 1,266
1 vote
0 answers
227 views

Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of $G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf ...
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2 votes
1 answer
120 views

Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram

A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...
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  • 21
6 votes
1 answer
306 views

CW structure on infinite-dimensional manifolds

It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they ...
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10 votes
1 answer
296 views

Simplicial simple homotopy vs. cellular simple homotopy

I recently started reading up on simple homotopy theory. Here is a question I stumbled upon. In his 1938 Paper Simplicial Spaces, Nuclei and m-Groups Whitehead introduced the notion of elementary ...
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3 votes
1 answer
250 views

$G$ uncountable implies $K(G,1)$ is not a finite CW complex

I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ ...
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13 votes
2 answers
759 views

Acyclic group and finite CW-complex

Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
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  • 707
2 votes
1 answer
311 views

CW complexes obtained by attaching cells not with increasing dimension

CW-complexes are defined by attaching cell with increasing dimension: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it ...
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  • 1,243
1 vote
1 answer
203 views

Fundamental group of the complement of cell subcomplexes

Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
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1 vote
0 answers
29 views

The cellularity of the composition of cellular maps (with arbitrary CW decompositioning)

Is the composition of cellular maps cellular? Related to this, I have another question. (I apologize to asking very similar question.) Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf ...
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  • 33
2 votes
1 answer
97 views

Is the composition of cellular maps cellular?

Let $X$, $Y$, $Z$ be topological spaces homeomorphic to CW complexes. And let $f:X\to Y$, $g:Y\to Z$ be cellular maps. My question is "Is the composition $g \circ f$ cellular map?". If $Y$ admits ...
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  • 33
4 votes
0 answers
171 views

Properties of triangulations of homeomorphic CW complexes

Let $X$ and $Y$ be two triangulable CW complexes which are homeomorphic. Is it true that there exists a triangulation of $X$ and a triangulation $Y$ which have a common subdivision?
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  • 180
3 votes
2 answers
276 views

Line bundles trivial outside of codimension 3

Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...
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  • 864
5 votes
1 answer
231 views

Curvature and asphericity of cube complexes

Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...
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