Questions tagged [cw-complexes]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
0 answers
140 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 ...
3 votes
1 answer
230 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 ...
12 votes
1 answer
1k 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-...
7 votes
2 answers
292 views

Homotopy domination of a wedge of two polyhedra

The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$. Question: Suppose that $...
1 vote
0 answers
124 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 ...
3 votes
0 answers
133 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 ...
4 votes
1 answer
436 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 $\...
12 votes
0 answers
255 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 ...
13 votes
0 answers
690 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 ...
5 votes
2 answers
423 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 ...
6 votes
2 answers
421 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 ...
1 vote
1 answer
220 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 $X'$. ...
0 votes
1 answer
334 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.
9 votes
1 answer
439 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 ...
10 votes
1 answer
414 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 ...
7 votes
1 answer
250 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}...
4 votes
0 answers
203 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-...
2 votes
0 answers
293 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 ...
2 votes
0 answers
291 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 ...
9 votes
0 answers
330 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 ...
5 votes
1 answer
550 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 ...
6 votes
1 answer
181 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 ...
1 vote
0 answers
432 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 ...
6 votes
1 answer
446 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 ...
30 votes
4 answers
3k views

Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...
3 votes
1 answer
316 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 $K(\mathbb{R},1)$ ...
13 votes
2 answers
855 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 ?
2 votes
1 answer
475 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 ...
4 votes
1 answer
644 views

Lifting cellular structures to fibrations, fibre bundles or coverings

It is a well known result in Algebraic Topology that given a covering space $E\to B$ where the base has a CW-structure, then the total space can be given a CW-structure (see for example Theorem 8.10 ...
1 vote
0 answers
30 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 ...
2 votes
1 answer
110 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 ...
4 votes
2 answers
871 views

A CW is of countable type, iff all its homotopy groups are countable? (References?)

When constructing a classifying space $BPL$ for piecewise linear microbundles, one would like it to be a polyhedron, i.e. a locally finite simplicial complex. Milnor solved this by showing that the ...
4 votes
0 answers
214 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?
7 votes
2 answers
1k views

homotopy pushout of spaces homotopic to finite CW complexes

Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW ...
3 votes
2 answers
293 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 ...
5 votes
1 answer
256 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). ...
5 votes
0 answers
243 views

Is my combinatorial set a CW complex?

I come from combinatorics and my notions of algebraic topology are very limited. I have a purely combinatorial definition of a certain set of "cells" and I want to know if what I have is "enough" to ...
4 votes
1 answer
247 views

Homotopy type of G-CW-structure

Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also? Edit: My main ...
2 votes
2 answers
477 views

Turning injection of homotopy groups to an isomorphism

Assume we have a connected CW-complex $Y$ and $X\hookrightarrow Y$ a connected sub-complex. We know that the inclusion induces injection on all homotopy groups. Is it true (or under what conditions it ...
3 votes
1 answer
154 views

A pair of spaces equivalent to a pair of CW-complexes

Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that $Z-A$ is homeomorhic to $X-Y$ and $Z/A$ homeomorphic to $X/Y$ and The closure of $Z-A$ ...
12 votes
0 answers
133 views

Finite list of neighborhoods to approximate any finite simplicial complex

It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...
3 votes
1 answer
305 views

Covering with Deck group $\mathfrak{S}_3$

I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...
4 votes
0 answers
421 views

Finite good covers on smooth manifolds

Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex. Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
4 votes
1 answer
244 views

Generalized cohomology of CW complex is direct limit?

Let $E$ be a (pre)spectrum (in the most classical sense, i. e. the sequence of CW complexes $E_n$ and maps $SE_n \to E_{n+1}$). Then we have the generalized cohomology theory $E^*$. For finite CW ...
1 vote
0 answers
149 views

Whitehead Theorem for maps

Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
4 votes
1 answer
381 views

null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes

$\require{AMScd}$ Preliminaries: Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is null-homologous (over $\mathbb{Z}_2$) if $...
3 votes
2 answers
509 views

Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
5 votes
2 answers
518 views

Second homotopy groups of 3-complexes and Fenn's spiders.

Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been ...
7 votes
2 answers
447 views

Critical points and high homotopy groups

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...
4 votes
0 answers
386 views

A cell decomposition of a CW-complex and, stratification of a topological space

What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ? I know that cell decomposition of a CW-complex is usefull to ...