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"Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$

Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices). I ...
M. Winter's user avatar
  • 12.8k
2 votes
0 answers
143 views

What is like "flat" but for based connected CW-complexes?

Let $X$ be a based connected CW-complex $X$. Say that $X$ is CW-flat if, for each map of based connected CW-complexes $f \colon Y \to X$ such that $\pi_n(f)$ is injective, $\pi_n X \wedge f$ is also ...
Ronald Z.'s user avatar
0 votes
1 answer
254 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 ...
William Thomas's user avatar
1 vote
0 answers
74 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 ...
brattok's user avatar
  • 11
9 votes
1 answer
370 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
  • 497
2 votes
0 answers
111 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
  • 141
5 votes
0 answers
82 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
114 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
633 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
92 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
102 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 ...
user avatar
3 votes
0 answers
173 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
1 vote
0 answers
52 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}\...
Mahtab's user avatar
  • 277
3 votes
1 answer
415 views

Spectral sequence in Adams'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
390 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
  • 277
0 votes
1 answer
224 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
  • 1,182
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 $f_i :X_{i+1} \to X_i$ ...
M.Ramana's user avatar
  • 1,182
0 votes
0 answers
86 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
213 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
260 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
  • 12.8k
6 votes
1 answer
412 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
275 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
190 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
140 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
159 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
217 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
  • 1,077
5 votes
0 answers
291 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
410 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
195 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
  • 1,182
1 vote
0 answers
271 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
  • 189
4 votes
2 answers
274 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
259 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
159 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
  • 1,069
1 vote
1 answer
197 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
4 votes
1 answer
270 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
  • 504
16 votes
2 answers
693 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
973 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
195 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
  • 767
3 votes
1 answer
211 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
211 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
119 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
235 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
  • 10.4k
1 vote
0 answers
213 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
  • 191
3 votes
0 answers
155 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
  • 5,861
11 votes
2 answers
665 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
  • 1,185
1 vote
3 answers
649 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
  • 1,069
6 votes
3 answers
846 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
  • 1,069
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
  • 3,554
3 votes
1 answer
242 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
  • 519
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