# Questions tagged [cw-complexes]

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96
questions

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### 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 ...

**7**

votes

**1**answer

187 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 ...

**3**

votes

**1**answer

186 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 $K(\mathbb{R},1)$ ...

**11**

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**2**answers

654 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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181 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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149 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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27 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

88 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 ...

**3**

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131 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?

**3**

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233 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

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191 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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157 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 ...

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205 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

285 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**

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143 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**

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115 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 $> 3$...

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252 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 ...

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207 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**

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**1**answer

166 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 ...

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121 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 ...

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202 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 $...

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288 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....

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327 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 ...

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180 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 ...

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191 views

### CW Product via Whitehead map

Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb ...

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529 views

### Allowing $G$-CW complexes to have more general cells

Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form ...

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282 views

### A weak version of the Whitehead Theorems

Let $f:X\longrightarrow Y$ be a map between CW-complexes $X$ and $Y$. By the Whitehead Theorems, if one of the conditions:
1- (homotopy version) $\pi_n (f):\pi_n (X)\longrightarrow \pi_n (Y)$ is an ...

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174 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 $...

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79 views

### Homeomorphism type of pair of faces in a regular CW complex

Let $X$ be a regular CW complex, $\sigma$ an $n$-dimensional cell of $X$ and $\tau$ an $(n-1)$-dimensional face of $\sigma$.
Is it true that the pair $(\bar\sigma, \bar\tau)$ is homeomorphic to the ...

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345 views

### Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes

Who was the first to prove this theorem and is there an "official" name for it?
Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on ...

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135 views

### 0-cells in CW complexes

If X is a CW complex, then for each fixed point x, is it possible adapt the cellular decomposution of X such that x be a 0-cell?
Actulally, my real interest: is any point in X nondegenerated?

**6**

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291 views

### How can I endow a “locally product” CW structure on a vector bundle over a CW complex?

I asked the same question in math stackexchange: https://math.stackexchange.com/questions/2322883/how-can-i-endow-a-locally-product-cw-structure-on-a-vector-bundle-over-a-cw-co
but it seems that it's ...

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227 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 ...

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782 views

### Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?

My question is
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
(My thoughts on this which might not be useful at all.) Since an ...

**5**

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**1**answer

673 views

### attaching maps in CW complexes

Suppose I have a finite CW complex $X$ with $p$-skeleton $X^{(p)}$.
Let $\varphi_f \colon S^p \to X^{(p)}$ be part of the attaching map of a $(p+1)$-cell $f$.
Let $\Phi_e \colon D^p \to X^{(p)}$ be ...

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178 views

### Existence of a perfect discrete Morse function

Let $X$ denote a regular cell structure on a closed (orientable) $n$-manifold (If it helps, the cells are polytopal and the attaching maps are affine).
Recall that a discrete Morse function on this ...

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453 views

### Is there an analogue of CW-complexes built from $K(\mathbb Z, n)$ instead of $S^n$?

The question is motivated by Eckmann-Hilton duality and certain flaws of the homotopy category of CW-complexes. Unfortunately, I do not know the formalism of model categories, so excuse me if it is a ...

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473 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 ...

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**0**answers

256 views

### Manifolds and CW-complexes

Let us consider a category $C$ formed by topological spaces and continuous functions (or by smooth manifolds and smooth functions). We consider the morphism category $C_{2}$. An object of $C_{2}$ is a ...

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326 views

### homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe:
$\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...

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312 views

### CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...

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155 views

### SImple homotopy type of a mapping cone

Consider two CW complexes A and B. Let f be a continuous map from A to B. Take a cone of f and denote it by Cone(f). If homology complex of Cone(f) is acyclic, one can identify homologies of A and B. ...

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333 views

### Regular CW complex arising from a Morse decomposition

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic ...

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288 views

### Factorization of a certain map through a CW-complex

Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...

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129 views

### Acyclicity of covering space

Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...

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325 views

### rational cohomology of symmetric groups

Let $\Sigma_k$ be the $k$-th symmetric group and $B\Sigma_k$ be its classifying space. How to prove:
for any $n\geq 1$ and the $n$-skeleton $sk_n (B\Sigma_k)$, there exists a finite dimensional $CW$-...

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220 views

### Does anyone know any applications of CW-complexes in graph theory?

As everyone knows :P, a graph is a CW-complex of dimension 1. Knowing that, are there any interesting results in graph theory that arise from working with CW-complexes? And more specifically, in ...

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236 views

### Different model structures on Top

There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure.
In the Mixed model structure $\mathsf{...

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618 views

### Multiplicative cohomology theories and smash products

In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says
However, G. W. ...

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316 views

### Universal covering and double cover functors

Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...