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 ...
brattok's user avatar
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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 $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
J.K.T.'s user avatar
  • 487
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 ...
Eric Ley's user avatar
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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 ...
Peter Kropholler's user avatar
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 ...
kelly maggs's user avatar
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 $\...
R. van Dobben de Bruyn's user avatar
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-...
Philippe Gaucher's user avatar
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 ...
Ronald J. Zallman's user avatar
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-...
William Thomas's user avatar
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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}\...
Mahtab's user avatar
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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 ...
T. Wildwolf's user avatar
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^{...
Mahtab's user avatar
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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 $\pi_n (\phi)$...
M.Ramana's user avatar
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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 $f_i :X_{i+1} \to X_i$ ...
M.Ramana's user avatar
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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 ...
user48975's user avatar
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 ...
Nicholas Proudfoot's user avatar
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 $cd\in E(K_3)$ ...
M. Winter's user avatar
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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 ...
Markus Zetto's user avatar
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 ...
gola vat's user avatar
  • 179
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 ...
gola vat's user avatar
  • 179
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})...
Faye3's user avatar
  • 317
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 ...
gola vat's user avatar
  • 179
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 "(...
B.Hueber's user avatar
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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 ...
Lennart Meier's user avatar
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 &...
Matt Zaremsky's user avatar
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,...
M.Ramana's user avatar
  • 1,172
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'...
blancket's user avatar
  • 161
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 ...
pfw's user avatar
  • 43
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 ...
Giulio Lo Monaco's user avatar
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 ...
piper1967's user avatar
  • 1,059
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 ...
Arshak Aivazian's user avatar
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 ...
Steve's user avatar
  • 494
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 $...
Arshak Aivazian's user avatar
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{...
Arshak Aivazian's user avatar
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{...
N.B.'s user avatar
  • 757
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 ...
user420620's user avatar
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 ...
Arshak Aivazian's user avatar
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 ...
Keen-ameteur's user avatar
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 (...
wonderich's user avatar
  • 10.3k
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 ...
Uncool's user avatar
  • 181
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 ...
user127776's user avatar
  • 5,821
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 (...
ThorbenK's user avatar
  • 1,175
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 ...
piper1967's user avatar
  • 1,059
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)=...
piper1967's user avatar
  • 1,059
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 ...
KhashF's user avatar
  • 2,588
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 ...
Nik Bren's user avatar
  • 499
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-...
Anthony's user avatar
  • 283
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 ...
Someone's user avatar
  • 265
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 ...
user101010's user avatar
  • 5,319
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 $\...
cellular's user avatar
  • 913