Questions tagged [cw-complexes]
The cw-complexes tag has no usage guidance.
165
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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 $\...
- 1,023
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votes
1
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954
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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 ...
- 17.5k
12
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0
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255
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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 ...
- 121
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699
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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 ...
- 632
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2
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425
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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 ...
- 909
6
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2
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427
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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 ...
- 40.2k
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1
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221
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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 $X'$.
...
- 1,970
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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.
9
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1
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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 ...
- 1,175
7
votes
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250
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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}...
- 85
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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-...
- 41
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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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293
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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 ...
- 494
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330
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Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
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loop space of a finite CW-complex
Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...
- 1,023
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Invariant neighborhood in a G-CW complex
Let $G$ be a discrete group and $X$ be a $G$-CW complex. For any $x\in X$ and open neighborhood $U$ of $x$, I am interested in the question that whether we can find a $G_x$-invariant open ...
- 411
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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 ...
- 1,596
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435
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Cellular chain complex of $G$-CW-complexes & their differentials
I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf
...
- 6,000
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1
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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 ...
- 21
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460
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CW structure on infinite-dimensional manifolds
It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they ...
10
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415
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Simplicial simple homotopy vs. cellular simple homotopy
I recently started reading up on simple homotopy theory. Here is a question I stumbled upon.
In his 1938 Paper Simplicial Spaces, Nuclei and m-Groups Whitehead introduced the notion of elementary ...
- 141
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$G$ uncountable implies $K(G,1)$ is not a finite CW complex
I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ ...
- 2,523
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857
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Acyclic group and finite CW-complex
Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
- 707
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1
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482
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CW complexes obtained by attaching cells not with increasing dimension
CW-complexes are defined by attaching cell with increasing dimension: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it ...
- 1,263
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1
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293
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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 ...
- 154
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30
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The cellularity of the composition of cellular maps (with arbitrary CW decompositioning)
Is the composition of cellular maps cellular?
Related to this, I have another question. (I apologize to asking very similar question.)
Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf ...
- 33
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110
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Is the composition of cellular maps cellular?
Let $X$, $Y$, $Z$ be topological spaces homeomorphic to CW complexes. And let $f:X\to Y$, $g:Y\to Z$ be cellular maps.
My question is "Is the composition $g \circ f$ cellular map?".
If $Y$ admits ...
- 33
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215
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Properties of triangulations of homeomorphic CW complexes
Let $X$ and $Y$ be two triangulable CW complexes which are homeomorphic.
Is it true that there exists a triangulation of $X$ and a triangulation $Y$ which have a common subdivision?
- 250
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293
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Line bundles trivial outside of codimension 3
Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...
- 914
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258
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Curvature and asphericity of cube complexes
Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...
- 1,017
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244
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Is my combinatorial set a CW complex?
I come from combinatorics and my notions of algebraic topology are very limited. I have a purely combinatorial definition of a certain set of "cells" and I want to know if what I have is "enough" to ...
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Homotopy type of G-CW-structure
Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also?
Edit: My main ...
- 663
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480
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Turning injection of homotopy groups to an isomorphism
Assume we have a connected CW-complex $Y$ and $X\hookrightarrow Y$ a connected sub-complex. We know that the inclusion induces injection on all homotopy groups. Is it true (or under what conditions it ...
- 5,851
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1
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A pair of spaces equivalent to a pair of CW-complexes
Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that
$Z-A$ is homeomorhic to $X-Y$ and
$Z/A$ homeomorphic to $X/Y$ and
The closure of $Z-A$ ...
- 1,023
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Finite list of neighborhoods to approximate any finite simplicial complex
It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...
- 1,832
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1
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305
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Covering with Deck group $\mathfrak{S}_3$
I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...
- 1,623
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423
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Finite good covers on smooth manifolds
Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.
Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
- 180
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1
answer
246
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Generalized cohomology of CW complex is direct limit?
Let $E$ be a (pre)spectrum (in the most classical sense, i. e. the sequence of CW complexes $E_n$ and maps $SE_n \to E_{n+1}$). Then we have the generalized cohomology theory $E^*$.
For finite CW ...
- 41
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150
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Whitehead Theorem for maps
Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
- 537
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382
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null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes
$\require{AMScd}$
Preliminaries: Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is null-homologous (over $\mathbb{Z}_2$) if $...
- 1,221
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510
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Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$
Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
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448
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Critical points and high homotopy groups
Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...
4
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393
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A cell decomposition of a CW-complex and, stratification of a topological space
What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ?
I know that cell decomposition of a CW-complex is usefull to ...
- 325
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334
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CW Product via Whitehead map
Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb ...
- 283
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797
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Allowing $G$-CW complexes to have more general cells
Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form ...
- 6,722
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360
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A weak version of the Whitehead Theorems
Let $f:X\longrightarrow Y$ be a map between CW-complexes $X$ and $Y$. By the Whitehead Theorems, if one of the conditions:
1- (homotopy version) $\pi_n (f):\pi_n (X)\longrightarrow \pi_n (Y)$ is an ...
- 1,172
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2
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292
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Homotopy domination of a wedge of two polyhedra
The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$.
Question: Suppose that $...
- 1,172
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0
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85
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Homeomorphism type of pair of faces in a regular CW complex
Let $X$ be a regular CW complex, $\sigma$ an $n$-dimensional cell of $X$ and $\tau$ an $(n-1)$-dimensional face of $\sigma$.
Is it true that the pair $(\bar\sigma, \bar\tau)$ is homeomorphic to the ...
- 143
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503
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Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes
Who was the first to prove this theorem and is there an "official" name for it?
Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on ...
- 1,727
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241
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0-cells in CW complexes
If X is a CW complex, then for each fixed point x, is it possible adapt the cellular decomposution of X such that x be a 0-cell?
Actulally, my real interest: is any point in X nondegenerated?
