Questions tagged [cw-complexes]

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"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
334 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
106 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
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 ...
5 votes
0 answers
78 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 ...
9 votes
1 answer
604 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
93 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-...
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 ...
1 vote
0 answers
50 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}\...
4 votes
1 answer
368 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^{...
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
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 $\pi_n (\phi)$...
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 ...
14 votes
1 answer
941 views

Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?

The question is inspired by an answer to The concept of Duality It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
5 votes
1 answer
203 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
352 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 ...
11 votes
3 answers
1k 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
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
156 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
146 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
273 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 ...
5 votes
0 answers
195 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 "(...
10 votes
1 answer
577 views

Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?

Let $M$ be a connected open topological $d$-manifold (without boundary). Whitehead showed that if $M$ has a PL structure, there exists a subcomplex of dimension $\leq d-1$ onto which $M$ deformation ...
1 vote
1 answer
291 views

Fundamental group of the complement of cell subcomplexes

Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
8 votes
1 answer
396 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
259 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
248 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
155 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
186 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 ...
7 votes
2 answers
946 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{...
16 votes
2 answers
683 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 $...
2 votes
1 answer
175 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
205 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
109 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 (...
9 votes
1 answer
827 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 ...
0 votes
0 answers
199 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 ...
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 ...
11 votes
2 answers
664 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 (...
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
1 answer
356 views

Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram

A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...