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2
votes
1answer
80 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 $X'$ ...
10
votes
2answers
399 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
128 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
283 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
498 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
291 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 ...
5
votes
2answers
231 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
504 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 ...
2
votes
1answer
161 views

Extend Homeomorphism to Uniformly Continuous Function

I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$. I'm trying to build a CW-complex with it, so I want a continuous function from the closed ball $\overline{B}_n$ to the closure ...
5
votes
0answers
112 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). ...
20
votes
4answers
920 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 $k=1$ ...
0
votes
0answers
65 views

A sufficient condition for attaching squares to a 1 skeleton so that the CW-complex is a 2 - manifold

Suppose we have a finite connected graph $G$, I want to add 2 -cells to $G$ so that the 2 cells have boundaries of length 4 (squares) and so that $G$ is the 1 skeleton of a surface (2-manifold) ...
3
votes
3answers
322 views

Conditions for a graph to be the 1- skeleton of a Surface

Given a connected finite graph G with degree at least 2 at each vertex, what are the conditions G needs to assume in order to attach 2-cells so that the CW- complex is a closed compact surface(2 - ...
7
votes
1answer
310 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 ...
21
votes
0answers
801 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 ...
3
votes
2answers
255 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 ...
6
votes
1answer
244 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 ...
4
votes
3answers
526 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 ...
15
votes
1answer
848 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?
17
votes
1answer
781 views

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 ...
1
vote
1answer
472 views

Definition of CW complexes

In Spanier's Algebraic Topology, he defined CW complexes assumed an additional strange condition: the cell must have the coherent topology with the characteristic map and the inclusion map of its ...
6
votes
3answers
929 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 ...
3
votes
2answers
703 views

The definition of a CW complex and related notions

In the appendix of Allen Hatcher's book "Algebraic Topology", a CW complex is defined to be a space iteratively constructed by attaching $n$-cells onto an $(n-1)$ skeleton. There is a more general ...
6
votes
1answer
435 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 ...
15
votes
2answers
612 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
votes
3answers
966 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 $\pi_2(X)\ne 1$?
3
votes
2answers
488 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
votes
1answer
279 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 ...
30
votes
6answers
3k views

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 ...
17
votes
0answers
858 views

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 ...
5
votes
2answers
318 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) ...
7
votes
2answers
691 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 ...
1
vote
2answers
463 views

Contractibility of CW subcomplexes

Let $Y$ be a connected CW-complex and $X\subset Y$ a connected CW-subcomplex. Suppose that each cell of $X$ is the boundary of a cell of $Y$. Is this enough to conclude that $X$ is contractible in $Y$ ...
6
votes
0answers
296 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 ...
18
votes
2answers
1k 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 ...
2
votes
1answer
435 views

characterization of cofibrations in CW-complexes with G-action

Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps. I am using the model ...