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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CW approximation: why does $S^{n+1}\to X$ qualify as an attaching map for attaching $S^{n+1}$ to another space $Z$?

So i am still trying to understand the general proof of the CW approximation. At one point in the proof we have the inductively build CW complex $Z^{n+1}$ together with a map $f:Z^{n+1} \to X$ such ...
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Question about isomorphism of induced homomorphism in the proof of CW approximation.

I am currently working through the proof of one exercise which is Let $(X,x_0)$ be a path-connected space. Show that there is a $CW$-Complex $(Z,z_0)$ together with a map $f\colon (Z,z_0)\to (X,...
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1answer
35 views

Relative homotopy groups $\pi_k (S^n, S^1)$

I'm an undergraduate student currently studying Algebraic topology. I've been struggling to find all relative homotopy groups $\pi_k (S^n, S^1)$ for $n\geq 3$, $k\leq n$. Here are my thoughts: If ...
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1answer
45 views

Fundamental group of $X$, a CW complex is isomorphic to the fundamental group of its 2-skeleton

I'm trying to show that if $X$ is a CW-complex, then $$ \pi_1(X) = \pi_1(X^2)$$ where $X^2$ is the 2-skeleton. I found the following proposition in Hatcher's book: Proposition 1.26. (a) If $Y$...
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1answer
33 views

Showing $i_*:\pi_k(X^n, x_0) \to \pi_k(X, x_0)$ is injective for $k \le n-1$

I've got a question about the solution of the following exercise: Let $X$ be a connected CW-complex and $X^n$ its $n$-skeleton (i.e. the subcomplex of all cells of dimension $n$ or less). Denote by ...
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1answer
51 views

Pulback of a map to a pushout / pullback of cells in a fibration

Given a fibration $f:X \to B$ of CW complexes, it makes sense to guess that the pullbacks of a cell of $B$ will be a cell for $X$. That is, let $B_p$ be the $p$-th skeleton of $B$ and $X_p$ the ...
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Why the category of CW-complexes is not stable?

I am studying homotopy theory and I would like to understand better what it means for a category to be stable. For instance, the book I'm studying says that "the categories CW∗ and CCh+ have very ...
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2answers
178 views

Is a sequential limit of homeomorphisms a homeomorphism?

Suppose that we have a diagram in the category Top of the following form $$ \cdots X_2 \to X_1 \to X_0=X $$ where the arrows are homeomorphisms. Is it true that that the natural morphism $lim_i X_i \...
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40 views

CW Wedge sum Cofiber sequence

I was reading this paper by Bousfield on the localization of spectra. On page 5, Lemma 1.13, there's a rather small curious technical detail on wedge sum. We have for a limit ordinal $\lambda,B_{\...
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16 views

A pragmatic way of calculating with local degree

I understand the technical definition of local degree in a theoretical sense via isomorphism of $H_n(U, U_i \backslash \{x_i\}) \cong H_n(S^n)$ etc. However, I'm a bit confused by how exactly this ...
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24 views

3-Sphere Minus 1-Skeleton

Construct $S^3$ as the double of a convex Euclidean polyhedron. This can be seen as a cell decomposition of $S^3$, with $V$ 0-cells, $E$ 1-cells and $F$ 2-cells (the number of vertices, edges and ...
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1answer
36 views

Compute homology groups

I'm not confident with this kind of problem, so I post my solution here to ask for checking it. I want to compute the homology groups of the space obtained from two copies of $\mathbb{R} P^2$ by ...
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1answer
55 views

Why product of spheres contains wedge of spheres as a CW-subcomplex?

We have $S^k \times S^l$ with cellular structure $e^0_1\times e^0_2,~ e^k\times e^0_2,~ e^0_1\times e^l,~ e^k\times e^l.$ Why do first three cells form $S^k \vee S^l$? I mean, if we assume $k<=l$, ...
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1answer
21 views

Fundamental group of wedge sum of CW complex

Consider two pointed CW complexes $(X,x_0),(Y,y_0)$ and their wedge sum $(X \vee Y,a)$ with $a$ the identification of the two base points. I want to give a sufficient condition under which we have $$...
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1answer
32 views

References for tubular neighborhoods of (CW or simplicial) complexes embedded in Euclidean space

The following is Fernando Muro's comment to this post on MathOverflow: Any countable group is a colimit of a sequence of finitely presented groups (indexed by the natural numbers). For each of ...
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1answer
38 views

Equivalence of singular and CW cup products in cohomology of a space $X$

Let $X$ be a topological space, and let $C^*(X)$ denote the singular cochains of $X$ (with integral coefficients). The cup product in singular cohomology is defined (in e.g. Hatcher) in the following ...
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1answer
40 views

A compact subspace of a CW complex is contained in a finite subcomplex

In Hatcher's Algebraic topology p.520 he gives the following proposition and proof about CW-complexes which I'll copy partially for clarity sake. Propostion A.1: A compact subspace of a CW complex is ...
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Subspace topology in CW complexes

A subset $U$ of a CW complex is open iff for any closed cell $C$, the set $U \cap C$ is relatively open in $C$. My question is, can one characterize the subspace topology a similar way? That is, (when)...
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If $X$ is a $CW-$complex. Are $C_*(X)$ and $C^{CW}_*(X)$ weakly equivalent?

If $X$ is a $CW-$complex and we denote by $C_*^{CW}(X)$ the chain complexe given by $H_n(X_n,X_{n-1})$ in degree $n$ can we construct a weak equivalence $C^{CW}_*(X) \rightarrow C_*(X)$? I know their ...
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43 views

Calculating the cochain complex of $S^2 \times S^4. $

Calculating the cochain complex of $S^2 \times S^4. $ My calculation: $C^6 = \mathbb{Z}$ $C^5 = 0$ $C^4 = \mathbb{Z}$ $C^3 = 0 $ $C^2 = \mathbb{Z}$ $C^1 = 0$ $C^0 = \mathbb{Z}$ Am I correct? ...
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56 views

If a CW Complex captures the topology of manifold, what captures its geometry?

Context: I work on a scientific computing code (mostly fluid mechanics) and I am trying to learn differential geometry, understand it and then mimic its basic structures into a numerical code. My ...
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Exercise 16.10. on pg.356 in “Modern Classical Homotopy Theory.”

The question is asking to show that it suffices to prove Theorem 16.2 for a path connected space $X \in \mathcal{T_*}.$ Here is Theorem 16.2: I believe that the answer of this question is ...
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38 views

CW-complex structure of $T^2 = S^1 \times S^1$ with $S^1 \times \{1\}$ collapsed to a point

Consider the $2$-torus $T^2 = S^1 \times S^1$ and consider the space $X = T^2/(S^1 \times \{1\})$, $T^2$ with $S^1 \times \{1\}$ collapsed to a point. What CW-complex structure does $X$ have and how ...
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42 views

Simple questions about a finite tree [closed]

Let $X$ be a finite tree (a contractible graph) which has at least one edge. There is a vertex of $X$ that meets only one edge of $X$. If we exclude the edge (and the vertex) in 1 from $X$, then $X$ ...
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1answer
56 views

Property of a space induced by fundamental group and covering space.

This problem is from my past Qual. Let $(X,Y)$ be a CW pair with both $X,Y$ connected and $x_0\in X$ a basepoint. Assume that inclusion induced homomorphism $\pi_1(Y,x_0)\to\pi_1(X,x_0)=G$ is ...
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2answers
57 views

Example of cw complexes quasi isomorphic but not homotopic.

From Corollary 4.33 of Hatcher (which is the corollary of Hurewicz Theorem) the map $f:X \to Y$ between simply connected CW complexes is a homotopy equivalence if $f$ induces a quasi-isomorphism $f_{\...
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1answer
35 views

Every simplicial map is cellular.

A simplicial map is by definition a map $f:K\to L$ between simplicial complexes that sends each simplex of $K$ to a simplex of $L$ by a linear map taking vertices to vertices. A cellular map is by ...
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If $g$ is a simplicial map of a finite simplicial complex $X$, then the diagonal of the matrix of $g_*:H_n(X^n,X^{n-1})\to (X^n,X^{n-1})$ is zero

Let $X$ be a finite simplical complex, and let $g:X\to X$ be a continuous simplicial map such that $g(\sigma)\cap \sigma=\emptyset$ for every simplex $\sigma $ in $X$. Then why is the diagonal of the ...
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37 views

Efficient computation of e.g. product and quotient of CW complexes

I'm interested in computing operations on CW complexes, particularly multiple operations successively, such as taking a product and then a quotient, as you might to compute the CW complex structure on ...
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1answer
36 views

Homology of CW Complex Mod Skeleton

I'm trying to figure out a general approach to computing the homology groups for a space $X/X^m$ where $X$ is a CW complex of dimension $n$, and $X^m$ is its $m$-skeleton where $m<n$. I'm aware ...
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About cell complexes in Rourke and Sanderson

Let $K,L$ be pl cell complexes and $f:|K| \to|L|$, a map that is affine linear on every cell of $K$. I need to show that $M = \{A \cap f^{-1}(B): A \in K, B\in L\}$ is a cell complex. Note that I use ...
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1answer
49 views

Functor from $\textbf{Top}$ to $\mathcal{H}$

Let $\mathcal{H}$ be the homotopy category of spaces; i.e. $\mathcal{H}$ has as its objects the CW complexes and as its morphisms the homotopy classes of maps between them. I'm trying to understand ...
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3answers
69 views

Counterexamples to Whitehead's Theorem for non-CW complexes

Whitehead's theorem states that if X,Y are CW complexes and $f:X\to Y$ induces an isomorphism $f_* : \pi_n(X) \to \pi_n(Y)$ for all $n$, then $f$ is a homotopy equivalence. Are there simple ...
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1answer
97 views

Beginner Questions on CW-Complexes

As a beginner, I am struggling a bit with CW-complexes. I'm reading Hatcher, chapter 0. So I want to pose a few questions that are almost embarrassing to me but I believe it is important to ask such ...
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42 views

How do I calculate the cellular boundary map for $S^2$ with standard CW complex structure?

I'm using Hatcher's Algebraic Topology, and he gives: Let's give $S^2$ the standard CW complex structure consisting of two $0$-cells: $\{e^0_1, e^0_2\}$, two $1$-cells: $\{e^1_E, e^1_W\}$, two $...
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1answer
45 views

Cup products in CW complexes (Hatcher, Example 3E.6)

Let $X$ be obtained from $S^2 \times S^2$ by attaching a $3$-cell to the second $S^2$ factor by a map $S^2 \to S^2$ of degree $2$. Then from cellular cohomology it follows that $H^*(X;\Bbb Z)$ ...
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2answers
58 views

Is this the right way to construct the adjunction space homotopy equivalent to $S^1$? If not, then how to do it?

Currently, I'm working on an example from a topology book which states that The sphere $S^n$ can be obtained by attaching an $n$-cell to a space with one point: $D^n\cup_f\{a\}$. Question: I ...
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1answer
70 views

Understanding Lens space in Hatcher's Algebraic Topology

Update: I put my understanding in the answer. If you find any mistakes, please let me know. Thanks for your time. This is from Hatcher's Algebraic Topology, Example 2.43: Lens Spaces, page 144--146....
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1answer
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Attaching $2$-cell to a circle

I get this problem from my past Qual. " For $n=1,2,\dots$ let $f_n\colon S^1\to S^1$ be the map $z\mapsto z^n$. Define $$X(n)=S^1\cup_{f_n} e^2$$ as the cell complex obtained by attaching a 2-...
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Explicit cell structre of Torus with 3 points identified as 1

I have this problem "Let $X$ be the space obtained from the torus $S^1\times S^1$ by identifying three distinct points to one point. Find an explicit cell structure on $X$." It is a well-...
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1answer
57 views

Why do we need $A,B$ contractible for $A\cup B$ to be contractible?

THis is problem 0.23 in Algebraic Topology of Hatcher "Show that a CW complex $X$ is contractible if $X$ is the union of two contractible subcomplexes $A$ and $B$ whose intersection is also ...
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1answer
42 views

Union of a sequence of increasing CW-complexes and Retract

I have updated my question based on the comments. I think I can solve the problem. But I hope you can help me verify it. I am reviewing Algebraic Topology to prepare for my Qual. I have this problem ...
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1answer
75 views

Postnikov tower of a CW complex is unique up to homotopy equivalence

The above is extracted from Hatcher's Algebraic Topology. I have understood that every connected CW complex has a Postnikov tower, but I can't see how Corollary 4.19 implies that such a tower is ...
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1answer
50 views

When a quotient map of topological graph is open?

I follow the definition of a topology on a graph, from wikipedia: A graph is a topological space which arises from a usual graph $G=(E,V)$ by replacing vertices by points and each edge $e=xy\in E$ by ...
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38 views

Equivalent condition for deformation retraction

I am currently reading a paper where the author needs to show that a map $i: X \hookrightarrow Y$ between 2 $CW$- complexes $X$,$Y$ (and $X$ is contained in $Y$) is a deformation retraction. He ...
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1answer
38 views

Question about faces of a simplex and relation to complexes

I'm reading Hatcher's book on algebraic topology, p103: Let $[v_0, \dots, v_n]$ be an $n$-simplex. A face of $[v_0, \dots, v_n]$ is the $(n-1)$-simplex obtained by deleting one vertex $v_i$ from the ...
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1answer
69 views

Show that this gluing of spaces produces the circle.

I'm trying to understand this example here: Help understanding CW-complex construction. So basically, we consider a point $x_0 \in \mathbb{R}$ and we consider the set $$X^1 := \{x_0\} \sqcup [0,1]/\...
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52 views

ordering and orientation for CW complexes

The vertices of each cell of a simplicial complex are often assumed to have an ordering, for example in these notes, this exercise, and this blog post. From the ordering of the vertices of a simplex $\...
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3answers
45 views

Proof that there are only finitely many $1$-cells that have a $0$-cell as a boundary point in a CW complex

Below is a lemma from John Lee's Introduction to Smooth Manifolds. Suppose $M$ is a $1$-manifold endowed with a regular CW decomposition. We want to show that every $0$-cell of $M$ is a boundary ...
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1answer
59 views

Hatcher's proof of Quillen plus construction

This is from Hatcher's Algebraic Topology. This Proposition says we can kill the fundamental group of an arbitrary connected CW complex with $H_1=0$, without affecting the homology groups. The proof ...

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