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7
votes
2answers
256 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
469 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
259 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
217 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
votes
2answers
396 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
138 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
105 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). ...
19
votes
4answers
823 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
294 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
271 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 ...
18
votes
0answers
710 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 ...
2
votes
2answers
237 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
225 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
497 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 ...
14
votes
1answer
785 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
742 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
430 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
823 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
659 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
404 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
594 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
943 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
473 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
271 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 ...
29
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 ...
16
votes
0answers
725 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
299 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
654 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
458 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 $Y$ ...
5
votes
0answers
288 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
423 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 ...
