Questions tagged [cw-complexes]
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152
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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 $\pi_n (\phi)$...
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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 $f_i :X_{i+1} \to X_i$ ...
- 1,108
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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 ...
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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 ...
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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 $cd\in E(K_3)$ ...
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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 ...
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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 ...
- 159
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93
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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 ...
- 159
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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})...
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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 ...
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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 "(...
- 701
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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 ...
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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 &...
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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,...
- 1,108
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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'...
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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 ...
- 43
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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 ...
- 1,043
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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 ...
- 1,019
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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 ...
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“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 ...
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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 $...
- 5,260
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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{...
- 5,260
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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{...
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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 ...
- 317
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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 ...
- 5,260
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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 ...
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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 (...
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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 ...
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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 ...
- 5,607
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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 (...
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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 ...
- 1,019
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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)=...
- 1,019
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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 ...
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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 ...
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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-...
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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 ...
- 265
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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 ...
- 5,299
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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 $\...
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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 ...
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Smooth dual cell structure
Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
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Covering image of a connected CW-complex need not be a CW-complex
This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
- 612
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Regular or h-regular CW-complex structure for the Poincaré homology sphere
I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré ...
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CW-presentation of configurations of points in plane and space
I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the ...
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can we take skeletons of covering maps to give new covering maps?
Let $X$ be an $n$-dimensional cell complex.
We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$.
Take the universal cover (or a general covering space) $\tilde X'$ of $X'$.
...
- 1,950
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How do I show that any finite-dimensional (absolute) CW-complex $X$ is locally contractible?
I know that it holds even if $X$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.
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K-theory on finite-dimensional (possibly not finite) CW complexes
I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
- 1,107
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The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex
Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
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What does go wrong in Cellular homology if one considers projective limits of celullar complexes instead of CW-complexes?
Consider a nice topological space $X$ (e.g. the 3-sphere) and consider inside a decreasing sequence of compact subsets $(K_n)_{n\in\mathbb N}$ such that $K_\infty:=\bigcap_{n\in \mathbb N} K_n$ is 0-...
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What topological spaces can be realized as cell complexes?
What are the topological spaces can be realized as cell complexes, up to homeomorphism? It seems for instance that all manifolds can be built from cell complexes. It is clear however that one can ...
- 494
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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 ...
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