# Questions tagged [cw-complexes]

The cw-complexes tag has no usage guidance.

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### What does actually being a CW-complex provide in algebraic topology?

From time to time, I pretend to be an algebraic topologist. But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious. One that came up recently is the ...

**33**

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**2**answers

2k views

### Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...

**30**

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**4**answers

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### 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 $k=1$ ...

**30**

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**1**answer

2k views

### Manifolds admitting CW-structure with single n-cell

Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented):
When does $M$ not admit (up to homotopy-type) a CW-structure with a single $n$-cell?
By classification of ...

**24**

votes

**1**answer

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### When is a compact topological 4-manifold a CW complex?

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...

**22**

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**2**answers

2k views

### CW complexes and paracompactness

It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two ...

**19**

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**1**answer

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### Can the Alexander horned sphere arise as a cell boundary in a finite CW-sphere?

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes ...

**18**

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**4**answers

2k views

### When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...

**18**

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421 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 ...

**15**

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**1**answer

1k views

### Is a finite CW complex minus a point still homotopy equivalent to a finite CW complex?

Let $X$ be a finite CW complex and $x_0$ a point in $X$.
My question is then just:
Is $X-\{x_0\}$ still homotopy equivalent to a finite CW complex?

**15**

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762 views

### Does the following condition imply the homotopy type of a wedge of spheres?

Let me preface this question by saying that I am not an algebraic topologist.
Motivation. I was looking with a colleague at the homotopy type of a family of posets and we were able to show using ...

**15**

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**2**answers

919 views

### “Economic” CW-structure for Eilenberg-MacLane spaces?

The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...

**15**

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**1**answer

525 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. ...

**15**

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**3**answers

1k views

### The second homotopy group of a simple CW-complex

Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?

**15**

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**1**answer

312 views

### Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...

**14**

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**2**answers

859 views

### Counterexamples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...

**10**

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**3**answers

595 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 ...

**9**

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**2**answers

262 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 ...

**8**

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**1**answer

802 views

### the homotopy type of the pointed loop space of a countable cw complex

I apologize in advance if this is too elementary for this forum. I have received some help but am still unsure about how to proceed. I am interested in a proof of the following result due to John ...

**7**

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**2**answers

354 views

### Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...

**7**

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**3**answers

793 views

### Dual cell structures on manifolds

Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $...

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**2**answers

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### 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 ...

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460 views

### A description of cellular boundary maps in terms of a Morse function

I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...

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410 views

### Does a finite suspension spectrum make a space finite?

Suppose that $X$ is a space whose suspension spectrum $\Sigma_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma_+^\infty(X)$ is (weakly) ...

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769 views

### All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...

**6**

votes

**1**answer

424 views

### Increasing union of contractible CW complexes

Let X be CW complex. I'm trying to prove (using Zorn's lemma) that there is maximal contractible subcomplex. Problem is that I'm not able to show that increasing union of contractible subcomplexes has ...

**6**

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**1**answer

255 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 ...

**6**

votes

**2**answers

401 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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**1**answer

996 views

### Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...

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268 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 ...

**6**

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169 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 $...

**6**

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**1**answer

341 views

### State of knowledge on the Commutative W-spaces which appear in “Model Categories of Diagram Spectra”

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ...

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**0**answers

137 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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331 views

### The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...

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582 views

### Alexander duality theorem for CW-complexes and stable homotopy theory

In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem
Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$
for finite CW-complexes with a "nice ...

**5**

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**1**answer

254 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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264 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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### CW complex and group action

This is a general question and any reference or related result will be extremely helpful.
Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...

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**1**answer

471 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 ...

**5**

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**1**answer

272 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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151 views

### In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ...

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296 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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291 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 $CW$-...

**4**

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290 views

### Attaching cells of different dimensions at once in a CW-complex

Let $X$ be a CW-complex and $X^m$ it's $m$-skeleton. I think that for any $n\geq 2$ and $1\leq r\leq n-1$ it should be possible to obtain $X^{n+r}$ directly from $X^n$ via a homotopy push-out
$$\...

**4**

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451 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 ...

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489 views

### Attaching cells of different dimensions at once in a CW-complex II

This question is related to Attaching cells of different dimensions at once in a CW-complex There, I didn't manage to formalize the idea I had in mind, and ended up with a question whose answer was ...

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**3**answers

691 views

### Constructing a simplicial set homology-equivalent to a given CW complex

I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW ...

**4**

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**1**answer

322 views

### Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions.
Is the infinity-groupoid of a finite ...

**4**

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**1**answer

254 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 ...

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**1**answer

146 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 $...