All Questions
20 questions
3
votes
0
answers
150
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 140
8
votes
1
answer
224
views
Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
- 13.6k
0
votes
1
answer
328
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
4
votes
1
answer
280
views
“Combinatorial” moves between cell complexes
EDITED:
A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves.
Is there a similar thing on finite cell complexes? That is, are ...
- 504
2
votes
0
answers
346
views
regular CW complex and incidence matrices
Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case.
If it's ...
- 504
9
votes
0
answers
333
views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
14
votes
2
answers
906
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 ?
- 717
5
votes
1
answer
280
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). ...
- 1,017
4
votes
1
answer
432
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 $...
- 1,231
4
votes
1
answer
745
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 ...
- 909
6
votes
1
answer
2k
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 ...
- 7,637
4
votes
2
answers
451
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 $CW$-...
- 519
4
votes
0
answers
221
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 ...
- 15.5k
7
votes
2
answers
1k
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 $\...
- 699
5
votes
0
answers
167
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). ...
- 54.6k
30
votes
4
answers
3k
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 $k=1$ ...
- 6,210
35
votes
1
answer
3k
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 ...
- 17.5k
19
votes
4
answers
4k
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 $n$-...
- 732
17
votes
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 $\pi_2(X)\ne 1$?
- 1,181
16
votes
3
answers
2k
views
When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
