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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Cellular Approximation for Homeomorphism on a Compact Surface.

In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows: Suppose $M$ is a ...
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First homology group of a closed non-orientable 2-manifold vía the cellular homology groups

Let $N_h$ be a closed non-orientable 2-manifold of genus $h\geq 1$. I am trying to compute the first homology groups $H_1(N_h)$. For do so, it is sufficient compute the cellular homology group $H_1^{...
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Obstruction Theory in Seifert Surfaces

Background: This is from Livingston unplished notes in knot concordance: Proposition 1.7.1 claims: Let $K$ be a knot in $S^3$. Then every Seifert surface $F$ for $K$ has a function $f:S^3-\nu(K)\to S^...
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What are some uses of non-locally finite CW complexes?

A CW complex can be non-locally finite. An example of one is obtained by attaching a disk by its boundary to each point of $(0, 1)$, together with the end-vertices $\{0\}$ and $\{1\}$. When proving ...
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Quotient of finite pointed CW complex is a CW complex

Let $X$ be a finite pointed CW complex (where the basepoint forms a $0$-cell) and $A$ be a CW subcomplex of $X$ containing its basepoint. I am trying to determine if $X/A$ is also a finite pointed CW ...
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On abelian spaces

By Allen Hatcher's book, a space is called abelian if it has trivial action of $\pi_1$ on all homotopy groups $\pi_n$, since when $n=1$ this is the condition that $\pi_1$ be abelian. My quesion is ...
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Two CW complexes with the same number of $n$-cells for each $n$ homeomorphic?

i am trying to understand Theorem 5.20 in Lee's Introduction to Topological manifolds I am interested in the uniqueness part. Does this mean that two CW complexes with the same number of $n$-cells ...
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CW structure on the product $X\times Y$ of two CW-complexes $X$ and $Y$

I do understand that the $k$ -cells of $X\times Y$ are $\{e^i_X\times e^{k-i}_Y\}$ where $e^i_X$ is an $i$-cell of $X$ and $e^{k-i}_Y$ is a $k-i$ cell of $Y$ . I have the attaching maps from the CW-...
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  • 708
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Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells?

Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells? Euler Characteristic of $\mathbb{R}P^2$ is $\chi(\mathbb{R}P^2)=1$ Since it has 1 0-Cell,1 1-Cell and 1 2-...
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Reidemeister Torsion Example with Lens Spaces

In Andrew Ranicki's notes on Reidemeister torsion (https://www.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf), he gives the following example with lens spaces: where I copied the beginning of the ...
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Retracting a punctured cell in a CW-complex

In the proof of the cellular approximation theorem, Hatcher use the facts that if $Y$ is a CW-complex and $\ast$ is a point in an open $k$-cell $e^k$, then $Y-\ast$ deformation retracts into $Y-\...
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Space obtained by attaching two $2$-cells to $S^1$

I am working on the following problem: Let $X$ be the CW complex obtained from $S^1$ by attaching two $2$-cells: one by a map of degree $2$, and one by a map of degree $3$. (a) Compute the homology ...
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$\tilde{H}_q(S^n)=0$ for all $q\neq n$

Definition. A reduced ordinary homology theory $\tilde{H}_*$ consists of functors $\tilde{H}_q$ from the homotopy category of nondegenerately based spaces to the category of Abelian groups that ...
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The open $n$-cells of a CW-complex are open in the CW-complex

I am trying to prove that if $X$ is a CW-complex and $e_\alpha^n$ is an open $n$-cell, then $e_\alpha^n$ must be open in $X$. For do so, lets recall the following definitions: Definition 1. Let $\{...
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  • 493
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turn cofibration to fibre bundle

Suppose $X$ a CW complex, Y its subcomplex, $T$ is any parameter space. If we have map: $Y \to X \to X/Y$, the first map is inclusion and the second map is collapsing map, if we apply all spaces with ...
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The dependence of the homotopy type on the attaching map

Suppose we have $\vee_{i=0}^k S^2$ the two skeleton and $h:S^3\to \vee_{i=0}^k S^2$ the attaching map to attach cell $e^4$. My question is, if two attaching map $h_1$ and $h_2$ are homotopy, can we ...
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Is a topological space that is homotopy equivalent to a CW-complex locally path-connected?

While describing Quillen’s $ + $-construction in his book Algebraic K-Theory, V. Srinivas assumes that his topological spaces are equivalent to CW-complexes and are path-connected. As universal ...
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homology of circle times n-sphere

I would like to compute $H_i(S^1\times S^n;\mathbb{Z})$ for $i\geq3$. I believe one approach is to use the Mayer-Vietoris sequence. If we take the admissible cover consisting of $S^1\times S^n\...
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Construct a 2-dimension CW-complex whose fundamental group is Z $\times Z/2

2-D CW-complex. If Z is a CW complex with one 0-cell. For simplicity, the $\pi_1(X)$ has presentation where the generators are the 1-cells and the relation come for the 2-cells. More precisely, each 1-...
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Contractibility and uniform contractibility of metric spaces not homotopy equivalent to CW-complexes

I'm looking for an example of a metric space $X$ that is not contractible but satisfies the following "uniform contractability" condition: for every $r>0$ there exists $s\geq r$ such that ...
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CW complex which is a union of contractible subcomplexes with contractible intersection is contractible

Im trying to show that if X is a CW complex, A,B $\subset $X are contractible sub complexes with contractible intersection and $X=A\cup B$, then X is contractible. We can say that X/A$\cap$B is ...
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$X$ is a deformation retract of its mapping cylinder

I am trying to show the following: If $f : X \to Y$ is a homotopy equivalence and moreover a cellular map between finite CW-complexes, then $X$ is a deformation retract of $M_f$ via a cellular map. ...
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2 votes
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CW complexes with cofibration morphisms

Let $CW$ be the category of CW complexes and $CW_{cof}$ be the wide subcategory whose objects are CW complexes and morphisms given by inclusions of subcomplexes (i.e. cofibrations in the model ...
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Twisting Products in Hatcher's Algebraic Topology

In Allen Hatcher's book Algebraic Topology, in several places is used the terminology 'twisted product'; eg on page 338: Among other things, fibrations allow one to describe, in theory at least, how ...
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Calculating the simplicial homology of a tetrahedron.

I want to calculate the simplicial homology of the $\Delta$-complex figure on the right: I believe that it consists of $2$ 0-simplices, four $1$-simplices and three $2$-simplices. So I am now in the ...
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What are the morphisms in the 'category of CW-pairs'?

A CW-pair consists of a CW complex $X$ (with cell decomposition $\mathcal{E}\equiv\{e_\alpha\}_{\alpha\in I}$) together with one of its subcomplexes $A$ (a closed subspace of $X$ consisting of a union ...
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A pair $(X,A)$ is $n$-connected iff the inclusion $A\rightarrow X$ is an $n$-equivalence.

This book says followings: Definition. A continuous map $e:A\rightarrow Z$ is an n-equivalence if, for all $y\in Y$, the induced map $e_{*}:\pi_q(Y,y)\rightarrow \pi_q(Z,e(y))$ is an injection for $q&...
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Is my understanding of CW complexes correct?

I'm very new to algebraic topology and am trying to wrap my head around CW complexes. My main sources are Allen Hatcher's Algebraic Topology and these online notes by Soren Hansen. I outline my ...
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The difference between the boundary maps of 2 tetrahedrons

Here is the first figure on pg.105 in Allen Hatcher: But I am not sure why the negative sign before $[v_0, v_2, v_3]$ and $[v_0, v_1, v_2]$ in the boundary map. Could someone explain this to me ...
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why the word "quotient" is stated in the question and what will happen if ordering of vertices is not preserved?

I am trying to solve the following question: What familiar space is the quotient $Delta$-complex of a $2$ simplex$(v_0, v_1, v_2)$ obtained by identifying the edges $[v_0, v_1]$ and $[v_1, v_2],$ ...
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Hatcher's algebraic topology text, Problem 0.23

I am working on Problem 0.23 in Hatcher's Algebraic Topology text. It is as follows: Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is ...
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7 votes
1 answer
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$\pi_1$-equivalence of CW-complexes

Definition. We call $\pi_1$-equivalence the closure of a binary relation between topological spaces "there is a continuous mapping inducing an isomorphism of fundamental groups" to an ...
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1 answer
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Why is this CW complex connected?

I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows: Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 ...
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2 votes
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Are CW complexes absolute neighbourhood extensors?

Are CW complexes absolute neighbourhood extensors? A topological space $K$ is an absolute neighbourhood extensor (ANE) if for every space $X$, closed subset $A\subseteq X$, and continuous map $f:A\...
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3 votes
1 answer
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Is every open topological $n$-manifold homeomorphic to a CW complex? $(n \geq 5)$

The question of whether a closed topological manifold admits a CW structure is well-understood (with positive answer) for $n \neq 4$, thanks to the work Quinn ($n=5$) and Kirby-Siebenmann $(n >5)$...
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4 votes
1 answer
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Every CW complex is homotopy equivalent to a CW complex with 1 0-cell

I want to prove that Every path connected CW complex $X$ is homotopy equivalent to a complex $Y$ with a single $0$-cell. Notice that I'm making no assumption on the dimension of $X$ or the number of ...
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1 vote
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Equivalence of homology theories via acyclic models

I'm looking for a proof of "any two homology theories are equivalent" (obviously with some other hypothesis) via the Acyclic Models Theorem. I know that this is an application of the Acyclic ...
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3 votes
1 answer
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Homotopy and homology groups of finite CW complex

If $X$ is a finite CW complex, then $H_n$ and $\pi_n$ are finitely generated for $n\geq 2$. I think there could be two possible interpretation of finite CW complex. First one is $X$ is $n$-...
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6 votes
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Cellular homology computation

I know the theory behind cellular homology, but I'm having troubles in the actual computation. The cellular boundary formula states that: $$d(e^n_{\alpha})=\sum_{\beta}d_{\alpha \beta}e^{n-1}_{\beta}$$...
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3 votes
1 answer
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Second group homotopy CW complex relative 1-skeleton

Let X be a connected complex CW and $X^1$ his 1-skeleton. I want to show that: $$ \pi_2(X,X^1) \cong \pi_2(X) \times \text{ker}(i_*)$$ where $i_*:\pi_1(X^1) \rightarrow \pi_1(X)$ is the induced ...
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2 votes
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Cellular map on a sub CW-complex defines CW-structure on the pushout of CW-complexs

I want to show that given a subcomplex $A$ of a CW-complex say $X$ and a cellular map $f\colon A \to Y$ where $Y$ is another CW complex, the pushout $X\cup_A Y$ inherits CW-structure. I have an idea ...
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CW structure of $S^n\times S^n$

I am recently studying cellular homology but I am not very familiar with the CW complex. To compute the homology of $S^n\times S^n$, we can check the CW structure of $S^n\times S^n$ that has one $0$-...
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Subset of a CW complex containing one point from each cell is a closed discrete subset.

Suppose that $K$ is a compact subset of a CW complex space $X$. If $K$ intersects infinitely many cells, by choosing one point of $K$ in each such cell we obtain an infinite subset of $K$, say $A$. In ...
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Fiber of natural map of adjuction space

I have been studying CW complexes and I don't think I really understand adjuction spaces. I have the following problem. Let $X, Y$ be Hausdorff spaces, $A \subseteq X$ compact and $f \colon A \to Y$ ...
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1 answer
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Map on homology induced by covering map $S^n \to \mathbb{R}P^n$

Given the standard covering map $p: S^n \to\mathbb{R}P^n $ and I want to find the map $p_{*}: H_n(S^n, R) \to H_n(\mathbb{R}P^n, R)$ (R is an arbitrary ring). I introduce the following CW structures ...
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show $X^{(0)}$ is a discrete closed subset of $X$

If $(X,\mathcal{C})$ is a CW complex, then $X^{(0)}$ is a discrete closed subset of $X$. Let $A\subset X^{0}$. For any cell $c\in\mathcal{C}$, let $X_c$ be a finite subcomplex that contains $\bar{c}$....
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Are function spaces between CW complexes delta-generated?

Let $X$ and $Y$ be CW complexes. I wish to consider the mapping space $\mathbf{map}(X,Y)$ of continuous maps $X\to Y$, equipped with the (compactly generated version of the) compact-open topology. I ...
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4 votes
1 answer
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Lemma 2.34 of Hatcher's Algebraic Topology

I'm having trouble understanding the proof of Lemma 2.34 in Hatcher's "Algebraic Topology". More specifically I don't understand the proof of the following statement if $k=n-1$: The ...
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What Percentage of Type F Groups are not Poincare Duality Groups?

What "percentage" (in a meaningful sense of the term) of Type F groups are not Poincare duality groups (over any coefficient ring, in particular, over $\mathbb{F}_2$)? Of course, it's an ...
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Dimension of homotopy equivalent CW complexes [closed]

Let $X$ and $Y$ be $n$ and $m$-dimensional CW complexes. If $X\simeq Y$ then $n =m$? I think it's true but I don't know the reason.
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