Questions tagged [cw-complexes]
The cw-complexes tag has no usage guidance.
1
vote
0answers
101 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
1answer
144 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
2answers
187 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....
0
votes
0answers
85 views
$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$
I have to prove that the following equality $$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$$ holds for a CW-pair $(X, A)$ and a fixed CW-complex $K$; here $\wedge$ denotes the smash product, defined as $...
6
votes
2answers
263 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 ...
3
votes
0answers
107 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 ...
3
votes
1answer
129 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 ...
18
votes
2answers
414 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 ...
5
votes
1answer
263 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 ...
6
votes
2answers
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 $...
1
vote
0answers
66 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 ...
5
votes
1answer
248 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 ...
2
votes
0answers
103 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
votes
1answer
254 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 ...
3
votes
0answers
122 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 ...
10
votes
3answers
586 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
votes
1answer
456 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 ...
6
votes
0answers
133 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 ...
6
votes
2answers
400 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 ...
3
votes
1answer
312 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 ...
3
votes
0answers
209 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 ...
2
votes
1answer
233 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) ...
4
votes
2answers
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 ...
3
votes
0answers
148 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. ...
9
votes
2answers
260 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 ...
5
votes
1answer
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(...
1
vote
0answers
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 ...
4
votes
2answers
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 $CW$-...
0
votes
0answers
176 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 ...
1
vote
1answer
219 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{...
15
votes
1answer
513 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. ...
4
votes
1answer
250 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 ...
14
votes
2answers
855 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 $q=n+1$, ...
4
votes
0answers
180 views
Contractibility of regular CW sphere minus open star
Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains ...
4
votes
1answer
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 ...
1
vote
0answers
109 views
Explicit calculation of G-CW(V) structure of a G-space
I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...
6
votes
1answer
989 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 $G$, $H$ ...
2
votes
0answers
37 views
Conditions for monotone function to take maximal chains to maximal chains surjectively
Suppose that $P$ and $Q$ are graded posets (with rank function $r$) and suppose that all maximal chains of $P$ and $Q$ have length $n$.
Let $f:P \to Q$ be a surjective monotone function such that $r(...
2
votes
1answer
247 views
Two-bridge knots and CW-complex
The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation.
On the other hand, it's possible to provide a CW-complex with only one 0-cell ...
33
votes
2answers
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 ...
7
votes
3answers
780 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 $...
15
votes
1answer
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 ...
3
votes
1answer
266 views
Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?
Let $X$ be a finite, regular CW complex, and let $X'$ be its barycentric subdivision (i.e. the order complex of the face poset of $X$). Assume $X$ is collapsible.
Is $X'$ collapsible?
Is $X'$ ...
15
votes
2answers
910 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 ...
4
votes
1answer
240 views
CW complex with generalized cells
In the definition of CW complexes, all cells are homeomorphic to closed balls.
I search for a generalized notion of CW complexes. In my application, the complexes are in fact finite.
Is there a ...
7
votes
2answers
350 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
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...
6
votes
2answers
766 views
All mapping space between CW complexes is a CW complex?
Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...
4
votes
2answers
488 views
Attaching cells of different dimensions at once in a CW-complex II
This question is related to Attaching cells of different dimensions at once in a CW-complex There, I didn't manage to formalize the idea I had in mind, and ended up with a question whose answer was ...
4
votes
2answers
289 views
Attaching cells of different dimensions at once in a CW-complex
Let $X$ be a CW-complex and $X^m$ it's $m$-skeleton. I think that for any $n\geq 2$ and $1\leq r\leq n-1$ it should be possible to obtain $X^{n+r}$ directly from $X^n$ via a homotopy push-out
$$\...
5
votes
2answers
1k views
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 ...
