Questions tagged [cw-complexes]
The cw-complexes tag has no usage guidance.
164
questions
1
vote
0
answers
58
views
"Star" of a CW-complex
Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$?
I don't know if it has a name, but a similar ...
8
votes
1
answer
320
views
Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?
Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
2
votes
0
answers
104
views
A cell complex constructed from singular knots
Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...
5
votes
0
answers
77
views
Replacing a $G$-CW-complex with a $G$-homotopy equivalent $G$-simplicial complex - can anyone supply a reference?
Let $G$ be a group (not a topological group, just a group). By a $G$-complex I mean a CW-complex with an action of $G$ that takes cells to cells so that the pointwise and setwise stabilizer of each ...
5
votes
1
answer
103
views
Minimal cell structures in combinatorial model categories
I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the ...
9
votes
1
answer
596
views
Homotopy groups of finite CW complex finitely generated as Lie algebra
This is probably a well-known question, but I haven't found the answer on MO or MSE.
It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
2
votes
0
answers
89
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
2
votes
0
answers
92
views
Differential graded modules and the Serre-Swan theorem
I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules.
If $E \rightarrow B$ is a vector bundle, or even just any ...
3
votes
0
answers
172
views
The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
1
vote
0
answers
48
views
The number of $n$-cells attaching to $K^{n-1}$ in Wall's construction
Let $\phi:K\to X$ be a map, with mapping cyliner $M=X\cup_{\phi}(K\times I)$. We define $\pi_n (f)$ as $\pi_n (M,K\times 1)$. An element of $\pi_n (f)$ is represented by a pair of maps $\beta :S^{n-1}\...
3
votes
1
answer
347
views
Spectral sequence in Adam's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
4
votes
1
answer
361
views
Homotopy groups of mapping cylinder
Let $f:K\to X$ be a map, with mapping cyliner $M=X\cup_{f}(K\times \{ 1\})$. We define $\pi_n (f)$ as $\pi_n (M,K\times \{ 1\})$. An element of $\pi_n (f)$ is represented by a pair of maps $\alpha :S^{...
0
votes
1
answer
213
views
Explaining some detail in Wall's paper of CW-complexes
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\cup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\phi)$...
1
vote
0
answers
177
views
Does this sequence stop?
Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
0
votes
0
answers
77
views
Cellular structure of BSU(n)
I read somewhere that $BSU(n)$ has a cellular decomposition that consists of one 4-cell and higher dimensional cells. Can someone tell me why this is the case? In fact I am not sure if this statement ...
5
votes
1
answer
201
views
The bounded complex of a polyhedral decomposition
Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
The union ...
4
votes
0
answers
259
views
Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?
Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
6
votes
1
answer
350
views
Exit path categories of regular CW complexes
Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...
5
votes
1
answer
269
views
Homology of spherical $3$-manifold group
I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true.
Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
2
votes
0
answers
151
views
Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group
Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...
4
votes
1
answer
135
views
CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})...
1
vote
0
answers
144
views
Second homology group of a presentation complex
I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...
5
votes
0
answers
192
views
Does the (Poincare) dual complex represent the same topology?
To start with, consider some abstract $3$-dimensional simplicial complex $\Delta$ representing a manifold without boundary, for simplicity. Then, there is this well-known construction of the "(...
5
votes
0
answers
263
views
CW-structure on flag manifolds
I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer.
Let $G$ be a compact Lie ...
8
votes
1
answer
392
views
Contractible subcomplex containing 1-skeleton?
Question: If $X$ is a simplicial complex that's simply connected and $2$-dimensional, does there always exist a contractible subcomplex $Y$ satisfying $X^{(1)} \subseteq Y$?
The statement is true &...
0
votes
1
answer
191
views
Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?
Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
1
vote
0
answers
256
views
The mapping cylinder of a map between spaces that are homotopy equivalent to CW complexes
Suppose $X$ and $Y$ are spaces that are homotopy equvialent to CW complexes, and let $f:X\to Y$ be a continuous map. I am trying to show that the pair $(M_f,X)$ is homotopy equivalent to a CW pair.
I'...
4
votes
2
answers
260
views
Special cell decomposition for spheres with free $\mathbb{Z}/p\mathbb{Z}$-action by orthogonal transformations?
Consider the unit sphere $S^d$ in $\mathbb{R}^{d+1}$ with the antipodal action $\nu \colon x\mapsto -x$. This turns $S^d$ into a free $\mathbb{Z}/2\mathbb{Z}$-space.
Construct a CW-complex structure ...
2
votes
0
answers
247
views
Non-equivalent spaces with the same homotopy groups
It is well known that two topological spaces that have all homotopy groups isomorphic need not be weakly homotopy equivalent, because it might not be possible to construct a single map inducing all ...
0
votes
0
answers
152
views
Presentation complex of a finite perfect group and its features
Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...
1
vote
1
answer
185
views
Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?
Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the ...
3
votes
1
answer
263
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 ...
16
votes
2
answers
678
views
Is the decomposition of the homotopy type of a complex into a bouquet unique?
Is it true that if $A_1 \vee A_2 \vee .. \vee A_n = B_1 \vee B_2 \vee .. \vee B_m$, where $A_i, B_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A_i$ and $...
7
votes
2
answers
945
views
Does there exist a complete algebraic invariant of the homotopy type of a finite CW-complex?
Let $\mathrm{Cell}$ be the homotopy category of finite cell complexes. The main motive of my question
Is it true that for any algebraic category $A$ there is no fully faithful functor $F: \mathrm{...
2
votes
1
answer
173
views
Double coset decomposition for compact Lie groups
The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows
\begin{...
3
votes
1
answer
204
views
Ehresmann's fibration theorem for CW or simplicial complexes
Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ?
Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is ...
3
votes
0
answers
205
views
CW-complexes that cannot be homotopically compressed
Definition. A CW-complex $A$ can be elementary compressed to a CW-complex $B$ if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according ...
3
votes
0
answers
108
views
Description of Anderson-Putnam CW-complex construction
I have been trying to read the paper, Topological invariants fo substitution tiling and their associated $C^*$-algebras, to learn more about a construction of Anderson-Putnam complexes. However, it ...
3
votes
0
answers
232
views
CW structure on $\mathrm{PU}(3)$/Heisenberg group
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where
$H$ is the Heisenberg group of order 27
$G_{81}$ is the No. 9 group of order 81 (...
0
votes
0
answers
198
views
Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
3
votes
0
answers
154
views
When is the space of maps between varieties a finite CW complex?
$\DeclareMathOperator\Cont{Cont}$Given two algebraic varieties over $\mathbb{C}$ denoted by $X$ and $Y$ where $Y$ is projective and $X$ is either projective or affine/Stein. The space of continuous ...
11
votes
2
answers
663
views
Can we embed a closed manifold into a homotopy equivalent CW complex?
Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...
1
vote
3
answers
644
views
How can I construct a closed manifold from a finite CW complex?
If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...
6
votes
3
answers
828
views
Finite CW complex with finite non-abelian fundamental group and higher homologies zero
I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$
From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=...
2
votes
0
answers
140
views
Dimension range for non-torsion homotopy groups
Is there a constant $c$ for which the following is true?
Let $X$ be a connected finite CW complex of dimension $d$. For any $i>cd$, the homotopy group $\pi_i(X)$ is torsion.
What if we replace ...
3
votes
1
answer
229
views
Embedding CW-complexes into infinite-dimensional topological vector spaces
Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
12
votes
1
answer
1k
views
Whitehead product and a homotopy group of a wedge sum
Note : this is a crosspost from the Mathematics StackExchange, as suggested by this meta post.
Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-...
1
vote
0
answers
123
views
Codimension one submanifold gives cofibrant pair
Let $M$ be a smooth manifold, and $N$ be an embedded smooth submanifold of $M$ with $\partial M=\varnothing=\partial N$. Suppose, $\dim M-\dim N=1$, and $N$ is a closed subset of $M$.
Does the ...
3
votes
0
answers
132
views
Null-homotopic cellular loops are elementary null-homotopic?
I've got a 2-dimensional cell complex $X$ and a cellular closed loop $l \subset X$ that I happen to know is null-homotopic in $X$.
There are some very simple sorts of homotopies of cellular loops (or ...
4
votes
1
answer
433
views
Cellular homology of the universal cover
Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$.
Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...