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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Inclusion of the n-th skeleton in CW-complex is an n-equivalence

I am struggling with a guided exercise in which I need to prove the inclusion of the n-th skeleton in a CW-complex is an n-equivalence (meaning the induced maps on homotopy groups are isomorphisms for ...
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41 views

Why is the weak topology defined on cell complexes equivalent to the topology defined by the quotient map?

Just for background, I have been reading from Hatcher's textbook where he defines a cell complex in the following way: (1) Take any discrete set $X^0;$ we will call each element in this set a $0\...
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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 ...
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Compute $\pi_2(S^2 \vee S^2)$

As far as I know, there are two ways to calculate higher homotopy groups. One way is if we have a fibration then we get a long exact sequence in homotopy. The other is if we know a space is $(n-1)$-...
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$X=S^2/{\sim}$ where any point on the equator is identified with its antipodal point. Compute $\pi_1(X)$ and $H_\ast(X)$

My instructor gave me an idea. He used Van Kampen's Theorem to calculate the fundamental group, with $A=B=\mathbb{R}P^2$ and $A \cap B=S^1$ (the equator). I know $\mathbb{R}P^2$ is the quotient space ...
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computing $\pi_1(\mathbb{R}P^2 \vee\mathbb{R}P^2)$ and $\pi_1(\mathbb{R}P^2 \times \mathbb{R}P^2)$

could someone please check my solution? I'm trying to learn algebraic topology on my own. First, I calculate the fundemantal group of $\mathbb{R}P^2$. We know that $\mathbb{Z}/2 \rightarrow S^2\...
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$\Delta$-complex representing the Borromean rings

Consider the space $X=\mathbf{R}^3\setminus (S^1\sqcup S^1\sqcup S^1)$, where the circles are arranged in form of the Borromean rings. Let $a, b, c\in H^1(X)$ be the cohomology classes corresponding ...
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Covering $\Bbb RP^\text{odd}\longrightarrow X$, what can be said about $X$?

I am looking for any argument related to the following fact, which may or may not be true. Let $f:\Bbb RP^n\longrightarrow X$ be a covering space, where $n\geq 2$. Then, $X=\Bbb RP^n$. Now, for $n=\...
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compute $\pi_{1}(X)$ with cell complex structure.

algebraic topology Hatcher exercise 1.2.7 Let $X$ be the quotient space of $S^{2}$ obtained by identifying the north and south poles to a single point. Put a cell complex structure on $X$ and use ...
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Can a CW complex have a subset of cells that is itself a CW complex, yet is not a subcomplex of the larger complex?

Let $(X, \mathcal{E})$ be a CW complex and $\mathcal{E'} \subseteq \mathcal{E}$ be a subset of cells such that $\mathcal{E'}$ is a CW decomposition of the space $X' = \bigcup_{e \in \mathcal{E'}} e \...
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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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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
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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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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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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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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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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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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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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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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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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
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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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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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40 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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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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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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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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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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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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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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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
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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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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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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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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
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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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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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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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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 want to ...
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1answer
73 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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116 views

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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