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### Manifolds and CW-complexes

Let us consider a category $C$ formed by topological spaces and continuous functions (or by smooth manifolds and smooth functions). We consider the morphism category $C_{2}$. An object of $C_{2}$ is a ...
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### homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe: $\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...
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### CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...
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### SImple homotopy type of a mapping cone

Consider two CW complexes A and B. Let f be a continuous map from A to B. Take a cone of f and denote it by Cone(f). If homology complex of Cone(f) is acyclic, one can identify homologies of A and B. ...
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### Regular CW complex arising from a Morse decomposition

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic ...
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### Multiplicative cohomology theories and smash products

In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says However, G. W. ...
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### Universal covering and double cover functors

Initially posted on MSE Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...
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### Counterexamples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...
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### Contractibility of regular CW sphere minus open star

Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains ...
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### Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions. Is the infinity-groupoid of a finite ...
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### Explicit calculation of G-CW(V) structure of a G-space

I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...
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### Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...
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### Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...
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### Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?

Let $X$ be a finite, regular CW complex, and let $X'$ be its barycentric subdivision (i.e. the order complex of the face poset of $X$). Assume $X$ is collapsible. Is $X'$ collapsible? Is $X'$ ...
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### “Economic” CW-structure for Eilenberg-MacLane spaces?

The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
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### CW complex with generalized cells

In the definition of CW complexes, all cells are homeomorphic to closed balls. I search for a generalized notion of CW complexes. In my application, the complexes are in fact finite. Is there a ...
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### CW complex and group action

This is a general question and any reference or related result will be extremely helpful. Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...
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I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$. I'm trying to build a CW-complex with it, so I want a continuous function from the closed ball $\overline{B}_n$ to the closure $\... 0answers 133 views ### In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal? Suppose that you have decomposed a manifold$M$into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ... 4answers 1k views ### Is the space of diffeomorphisms homotopy equivalent to a CW-complex? Clarification: My question concerns the homotopy type of the space of$C^k$diffeomorphisms with the compact-open$C^k$topology, where$0< k \leq\infty$. I have stated my question below with$k=1$... 0answers 66 views ### A sufficient condition for attaching squares to a 1 skeleton so that the CW-complex is a 2 - manifold Suppose we have a finite connected graph$G$, I want to add 2 -cells to$G$so that the 2 cells have boundaries of length 4 (squares) and so that$G$is the 1 skeleton of a surface (2-manifold) ... 3answers 402 views ### Conditions for a graph to be the 1- skeleton of a Surface Given a connected finite graph G with degree at least 2 at each vertex, what are the conditions G needs to assume in order to attach 2-cells so that the CW- complex is a closed compact surface(2 - ... 1answer 370 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 ... 0answers 1k views ### Manifolds admitting CW-structure with single n-cell Let$M$be a topological$n$-manifold, closed and connected (not necessarily oriented): When does$M$not admit (up to homotopy-type) a CW-structure with a single$n$-cell? By classification of ... 2answers 328 views ### Second homotopy groups of 3-complexes and Fenn's spiders. Let$X$be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been ... 1answer 307 views ### State of knowledge on the Commutative W-spaces which appear in “Model Categories of Diagram Spectra” This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ... 3answers 607 views ### Constructing a simplicial set homology-equivalent to a given CW complex I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW ... 1answer 966 views ### Is a finite CW complex minus a point still homotopy equivalent to a finite CW complex? Let$X$be a finite CW complex and$x_0$a point in$X$. My question is then just: Is$X-\{x_0\}$still homotopy equivalent to a finite CW complex? 1answer 1k views ### Can the Alexander horned sphere arise as a cell boundary in a finite CW-sphere? Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes ... 1answer 568 views ### Definition of CW complexes In Spanier's Algebraic Topology, he defined CW complexes assumed an additional strange condition: the cell must have the coherent topology with the characteristic map and the inclusion map of its ... 4answers 2k views ### When is a finite cw-complex a compact topological manifold? I think the statement of the question is pretty straightforward. Given a finite$n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact$n$-... 2answers 969 views ### The definition of a CW complex and related notions In the appendix of Allen Hatcher's book "Algebraic Topology", a CW complex is defined to be a space iteratively constructed by attaching$n$-cells onto an$(n-1)$skeleton. There is a more general ... 1answer 565 views ### the homotopy type of the pointed loop space of a countable cw complex I apologize in advance if this is too elementary for this forum. I have received some help but am still unsure about how to proceed. I am interested in a proof of the following result due to John ... 2answers 696 views ### Does the following condition imply the homotopy type of a wedge of spheres? Let me preface this question by saying that I am not an algebraic topologist. Motivation. I was looking with a colleague at the homotopy type of a family of posets and we were able to show using ... 3answers 1k views ### The second homotopy group of a simple CW-complex Let$X$be a CW-complex with one 0-cell two 1-cells three 2-cells no cells in dimensions 3 or higher. Is it always true that$\pi_2(X)\ne 1\$?
In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem Theorem:[Alexander Duality] $$H^r(X,G)=H_{n-r+1}(S^n-X,G)$$ for finite CW-complexes with a "nice ...