Questions tagged [cw-complexes]

For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

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28 views

Connected sum of handlebodies has the homotopy type of a $1$-dimensional CW-complex

Let $M^3$ and $N^3$ be two compact connected $3$-dimensional handlebodies. It is easy to see that they have the same homotopy type of $1$-dimensional CW complexes. Is it true that their connected sum $...
15
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1answer
149 views

What manifolds $M$ have a $CW-$structure so that the $n-$skeleton, $M_n$, is a manifold for all $n$ aswell?

If you have a $CW-$structure on a connected manifold $M$ we obtain a filtration $M_n$ of $M$ where $M_n$ is the $n-$skeleton. If in addition we have that the $M_n$ are also all manifolds (like with ...
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1answer
45 views

Homotopy equivalence between pointed CW-complexes

Let $f : (X, x_0) \to (Y, y_0)$ a morphism in the category of pointed, connected CW-complexes. Let $A= [1/2, 1]^{+} \wedge X \cup Y \subset C_f$, where $C_f$ is the mapping cone and $X, Y$ are pointed,...
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1answer
81 views

Does this functor commute with inverse limits?

Let $F$ a contravariant functor from the category $A$ of pointed connected CW-complexes up to homotopy to the category $B$ of pointed sets, with $F$ sending coproducts of $A$ to products of $B$. Let $...
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1answer
73 views

Telescope of CW-complexes

Let $X_n$ the n-th term of an ascending sequence of pointed CW-complexes in their homotopy category. Let $X’= \bigcup_{n\geq 1} [n-1, n]^{+} \wedge X_n$ and let $A_1 = \bigcup_{k\geq 1, k odd} [k-1, k]...
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1answer
61 views

Equivalent versions of the Mayer-Vietoris axiom in Brown theorem

In the hypotheses of Brown representability theorem there is a contravariant functor F from pointed connected CW complexes to pointed sets, which must respect two axioms, the second of which is the so-...
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2answers
122 views

Does every finite $CW$ complex have the homotopy type of a smooth manifold? How about infinite $CW$ complexes?

Let M be a smooth manifold. I was wondering if every finite or infinite CW complex has the homotopy type of a smooth manifold because it seems way easier to compute De Rham cohomology groups than ...
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1answer
47 views

Isomorphism between homotopy groups of CW-complexes

Let $(Y, y_0)$ and $(Y’, y_0)$ pointed CW-complexes, with $Y’$ obtained from $Y$ by attaching $n+1$-cells. Why is it true that $i_{*}: \pi_{q}(Y, y_0) \to \pi_{q}(Y’, y_0)$ is an isomorphism for $q &...
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80 views

Proposition A.1 in Hatcher's AT, the last sentence

I have a problem with reading the proof of the proposition A.1 in Hatcher's Algebraic Topology book. In the last sentence he says $A \cup e^n_\alpha$ is a finite subcomplex containing $e^n_\alpha$. To ...
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1answer
87 views

Strong deformation retraction between CW-complexes, Brown theorem proof

I am currently reading Lemma 9.11 on Switzer, in the chapter dedicated to Brown representability theorem. However I am stuck on a few points of the proof. Let $F$ a contravariant functor from pointed ...
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1answer
63 views

Hatcher, topology of cell complexes

In the appendix to Algebraic Topology, Hatcher briefly goes over why A subset $A$ of a CW complex $X$ is open in $X$ $\iff$ $\Phi_\alpha^{-1}(A)$ is open in $D_\alpha^n$ for all $\Phi_\alpha$, where $...
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29 views

An example of a product of CW complexes that is not a CW complex

So I know that the product of two CW structures X and Y, one of which is locally compact, admits a CW structure. But I can’t think of an example when it fails. Is there a good example for this ...
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1answer
42 views

Regarding a space X as a CW complex

(My notations and definitions are similar to those in Hatcher's book.) In Hatcher's AT, we construct a CW complex in an inductive way. This creates a new topological space. Why don't we do a converse ...
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1answer
40 views

Proving that for a CW pair $(X,A)$, $X \times I$ deformation retracts to $(X \times \left\{ 0 \right\}) \cup (A \times [0,1])$

I am trying to understand the proof of Allen Hatcher's Proposition 0.16 from "Algebraic Topology". If $X$ is a cell complex, he defines $(X,A)$ to be a CW-pair if $A \subseteq X$ is closed ...
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1answer
75 views

Constructing simply connected CW complex with prescribed homology groups

I'm stuck on a past qualifying exam problem from my university: Construct two homotopically not equivalent simply connected CW complexes such that $H_0=\mathbb{Z}$, $H_2=\mathbb{Z}$, $H_3=\mathbb{Z}_2$...
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0answers
43 views

Terminology or notation for special CW-complex constructions

To compute homotopy groups of one-point unions of based spaces (for instance, spheres) it is relevant to consider quotient spaces $\prod_{i=1}^{n}X_i\Big/ \bigvee_{i=1}^{n}X_i$ where $X_1,X_2,\dots,...
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1answer
54 views

Homology of the unit ball $D^3$ with identification of boundary points by $180^\circ$ degree rotation around the vertical axis

This problem comes from Topology and Geometry of Bredon : Let $X$ result from $D^3$ by identifying points on its boundary $S^2$ taken into one another by the $180^\circ$ rotation about the vertical ...
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188 views

Examples of Finite-Dimensional Space with Non-Vanishing Homology in Higher Dimensions?

The Barratt-Milnor Sphere $X_n$ is an $n$-dimensional space which has non-vanishing singular homology in arbitrarily high dimensions. The space $X_n$ is a generalized Hawaiian Earring, i.e. the $n$-...
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55 views

Homotopy Equivalence of CW complex with maps of varying degrees.

I'm reviewing for an upcoming algebraic topology exam, and I have a question that I cannot solve fully. For $(m,n)\in \mathbb{Z}^2\setminus\{(0,0)\}$, let $X_{m,n}$ be a the CW complex obtained from $...
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1answer
57 views

Are pointed CW complexes for which the Yoneda embedding restricted to finite CW complexes reflects isomorphisms forced to be connected?

I'm currently reading Edgar H. Brown's paper, Abstract Homotopy Theory, which proves a categorical version of the Brown Representability Theorem. I've come across a statement that I don't really how ...
5
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1answer
86 views

Homology of Identifications of 2-torus

While working through old algebraic topology quals, I found these questions and wasn't sure if I had the right idea. Let $X$ be a closed genus–2 surface. Let $\alpha$ be a non-separating circle in $X$ ...
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1answer
287 views

Covering image of a connected CW-complex need not be a CW-complex.

Problem: Let $X$ be a connected CW-complex, and $Y$ be a connected topological space. Suppose $p: X\to Y$ is a covering map. Does there exist a CW-structure on $Y$? More generally, is $Y$ ...
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1answer
79 views

Two $\Bbb S^2$ with points identified [closed]

Suppose I have two 2-spheres $S_1$ and $S_2$ and $p_1, q_1\in S_1$ and $p_2, q_2\in S_2$. Then I identify $p_1$ to $q_1$ and $p_2$ to $q_2$. Let X denote this space. I want to study CW structure on ...
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34 views

CW complex from the fundamental polygon

Is there any way to get the CW complex structure from the fundamental polygon.I am having problem in finding the CW complex structure of objects which cant be visualized?
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1answer
112 views

How torsion arise in homotopy groups of spheres?

The example I have in mind of a torsion element in fundamental group is what happens on the projective plane: there is a cell $D^2$, but its boundary doubly covers a copy of $S^1$. The latter has then ...
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1answer
99 views

Suspension of a CW complex is a CW complex

How can I show that a suspension of a CW complex is a CW complex? I tried to start with showing first that a product of CW complexes is a CW complex but I didn't know how to construct the attaching ...
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1answer
155 views

Any two non-separating curves on a surface are equivalent

Problem: Let $\Sigma$ be a orientable surface possibly non-compact with boundary. Let $C,$ and $D$ be two simple-closed curves(smooth embedding) on $\Sigma\backslash \partial \Sigma$ such that both $\...
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37 views

Constructing a CW complex with following homology groups

Construct a CW Complex $X$ with the following homology groups (in coefficiants $\mathbb{Z})$: $H_0(X)=\mathbb{Z}, H_1(X)=\mathbb{Z}\oplus\mathbb{Z_2}, H_2(X)=\mathbb{Z_3}, H_3(X)=\mathbb{Z}, \; $ and ...
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1answer
126 views

Gluing of Möbius strips

Let $X$ be a Möbius strip. Take $n$ copies of $X$ and glue them by their boundaries (where the morphisms $S^1 \to S^1$ are homeomorphisms) and denote the new space by $Y$. What is a homotopy type of $...
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1answer
92 views

Identifying the differentials of the cellular chain complex of a (fat) geometric realization

Let $X_\bullet$ be a simplicial set, $\|X_\bullet\|$ its fat geometric realization and $|X_\bullet|$ its geometric realization, see here for definitions. The fat geometric realization $\|X_\bullet\|$ ...
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1answer
39 views

Two proper maps inducing same map at the level of fundamental groups are properly homotopic

I am dealing with the following problem: Let $X$ and $Y$ be two connected non-compact surfaces, possibly with boundary. Let $f_0,f_1:(X,x_0)\to (Y,y_0)$ be two proper maps with $f_{0*}=f_{1*}:\pi_1(X,...
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0answers
19 views

Intersection of cell with subcomplex

if I have a CW-complex $K$ and a subcomplex $L$, what can I say about an $n$-cell $K_{\sigma}$ if its intersection with $L$ is non-empty and not contained in the $(n-1)$-skeleton $K^{(n-1)}$? Does $K_{...
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0answers
44 views

constructing CW complex

While trying to constructing a CW complex of Moebius band, I come up an construction which I am not sure about it. The construction is as follows: Attach two 1-cells to a 0-cell. Then attached a 2-...
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1answer
45 views

$n$-skeleton in the definition of CW complexes

I try to understand the abstract definition of CW complexes in Hatcher's Algebraic Topology. Specifically, I refer to the following definition in the appendix of Hatcher's book. A CW complex is a ...
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0answers
44 views

Attaching disks and contracting is equivalent to taking quotients.

Take a Mobius band $M$ and attached a disk $D^2$ along its boundary, which is a copy of $S^1$ embedded in $M$. It is known that what we obtain is something homeomorphic to the projective plane. Now ...
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1answer
53 views

Basic Homotopy Question

I am starting to read the book "Rational Homotopy Theory" by Yves Felix, Stephen Halperin, J.-C. Thomas and I have a quick question about the very beginning (which only concerns basic ...
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0answers
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Different metrics on Cayley graphs

On every (say undirected, for simplicity) Cayley graph $\Gamma(G, S)$ we have the word ("geodesic") metric, that is $d_w(g, h)$ is the minimum of lenghts of paths joining $g$ to $h$. This ...
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1answer
46 views

K(G,1) space is unique up to homotopy equivalence

I am reading Hatcher's Proposition 1B.9 page 90. He is trying to prove that if $X$ is a connected CW complex, $Y$ is $K(G,1)$ then every homomorphism $\pi_1(X,x_0) \rightarrow \pi_1(Y,y_0)$ is induced ...
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0answers
25 views

$X$ is finite connected $n$-dimensional CW complex and $A = M_k(C(X))$.

Suppose there are positive $a,b,c\in A$ with $a,b,c\leq 1$ and $ab=b$ and $bc=c$, and $\text{rank}(c(t))\geq 2n+1$ at any $t\in X$. I want to prove that there is a nonzero projection $p$ such that $ap=...
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0answers
36 views

Recovering CW complex from a quotient

Let $X,Y,A,B$ be CW-complexes such that $A\subseteq X$ and $B\subseteq Y$ are subcomplexes. Assume that $(X,A)$ and $(Y,B)$ are good pairs if necessary. Suppose there is a relative homeomorphism $f:(X,...
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41 views

Why is this diagram in the quotient space $X/Y$ of CW-complexes a pushout square?

Suppose $X$ is an absolute CW-complex and $Y\subset X$ is a CW-subcomplex. I want to define a CW-complex for the quotient $X/Y$. I define it to be the sequence of spaces $π(X_n ∪ Y)$, where $X_n$ is ...
13
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0answers
333 views

Whitehead product and a homotopy group of a wedge sum

Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove the following isomorphism : $$\pi_{n+k+1}(X\vee Y)\cong\pi_{n+k+1}(X)\...
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0answers
139 views

Cellular chain complex of a CW-subcomplex $X'\subseteq X$

Let $X$ be a (possibly infinite) CW-complex, with a CW-subcomplex $X'$. The cellular chain complex of $X$, with coefficients in an abelian group $A$, is given by $\tilde{C}_n(X;A)=H_n(X_n,X_{n-1};A)$. ...
2
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1answer
133 views

Definition of Cell Decomposition?

In Chapter 5 of Lee's Intro to Topological Manifolds (page 130), he defines a cell decomposition as follows: I've been struggling to properly unpack this characterization. I have the two following ...
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1answer
27 views

Is the inclusion of the wedge sum into the reduced cylinder a relative cell complex?

For a CW-complex $X$ the map $$X\coprod X \hookrightarrow X \times I$$ is a relative cell complex. What I want to know is if this still holds in the pointed case. That is, if $X$ is a pointed CW-...
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1answer
41 views

CW Structure and connectedness.

Suppose $X$ be a $k$-connected CW-complex. I want to know Is it possible to find the value of $k$ from the CW-structure? In other words, Suppose $X$ has $n_i$, $i$-cell. What is the maximum value of $...
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0answers
26 views

Is ( SU(2n), Sp(n)) a CW pair?

I know that SO(n) can be given a CW complex structure, I wish to know if the pair (SU(2n), Sp(n)) is a CW pair?
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67 views

Question on tracial topological rank of C*-algebra and finite CW complex.

Denote by $I^{(0)}$ the class of finite dimensional C*-algebras and by $I^{(k)}$ the class of unital C*-algebras which are unital hereditary C*-subalgebras of C*-algebras of the form $C(X)\otimes F$ ...
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1answer
54 views

Are cells closed in CW Complexes and proving finite CW complexes are compact

So I am trying to understand the topology on cw complexes. I know that the $n+1$ skeleton of X is formed as $$X^{n+1} = \{ X^n \sqcup D_{i}^{n+1} \}/ \sim$$ where $\sim$ denotes $s \sim f_i^{n+1}(s)$ ...
2
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2answers
69 views

Smash product of CW complexes

I'm studying algebraic topology and I'm using Hatcher's book. There, he talks about the smash product of CW-complexes: Given two CW-complexes $X$ and $Y$ and two points $x_0 \in X$ and $y_0 \in Y$, ...

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