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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Compact subset of CW complex

Let $X$ be a CW complex. Prove that a set $A\subseteq X$ is compact if and only if $A$ is closed and has a nonempty intersection with finitely many open cells of $X$ only. What I know: compact means ...
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2answers
889 views

CW complex for Möbius strip

I was asked to find a CW complex for the Möbius strip with one 0-cell, two 1-cells, and one 2-cell. I can find a CW complex for a Möbius strip with more cells (two 0-cells, three 1-cells and a single ...
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1answer
105 views

CW complex definition ambiguity

I've seen two restrictions for a CW complex, and I am not sure which is right or if they are equivalent. A space $X$ is a CW complex if it has a partition into open sets $e^n_i$ such that: $X^n := \...
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472 views

CW complex with fundamental group $\Bbb Z/n$

I'm just learning about CW complexes. I came across this answer: "One way of constructing a connected $2$-dimensional CW complex with fundamental group $\Bbb Z/n$ (for some integer $n\geq 2)$ is to ...
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0answers
68 views

Showing that a CW complex is compactly generated

Let $X$ be a CW complex. I need to show that $X$ is finitely generated, which means that it's topology is coherent with its compact subspaces. A space $X$ is said to be compactly generated if: A ...
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0answers
288 views

Computing cellular homology of the sphere using a different CW structure

I want to compute cellular homology of the sphere $S^n$ using the CW structure where I have two cells $e^k_1,e^k_2$ for each dimension $0 \leq k \leq n$. Let me define the attaching maps to all be ...
2
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1answer
57 views

Give an example of an open cell in a cell complex for which the image of the characteristic map is not a closed cell.

Let $(X, \Gamma)$ be a cell complex, and let $e \in \Gamma$ be a open cell of dimension $n \geq 1$. I want to try and construct an example of a characteristic map $\Phi : D \to X$ (where $D$ is some ...
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2answers
74 views

Are all cell decompositions useful?

Based on the definitions and conventions used in the book Introduction to Topological Manifolds by John Lee, I can do the following. Let $X$ be a Hausdorff space, then define $\Gamma = \{\{x\} \ | \ ...
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3answers
211 views

Universal covering space of $\mathbb{R^3}\setminus S^1$

I am having a problems with describing the universal covering space of $\mathbb{R^3}\setminus S^1$. Namely, I need to find a CW-space which it is homotopy equivalent to. Well, I don't know how to ...
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2answers
65 views

If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals its closure

If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ I'm trying to prove the above which is a statement in Introduction to ...
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0answers
98 views

CW complexes are compactly generated

I am looking for verification and/or criticism of a proof I have constructed. The problem is to show that every CW complex is compactly generated. A topological space $X$ is $compactly$ $generated$ if ...
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44 views

CW complex, but with holes in cells

A CW complex is a specific kind of a partition of a topological space by open cells where each open n-cell is homeomorphic to an open n-ball. Consider a modified concept where each open n-cell is ...
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93 views

CW complex infinite-dimensional

Lemma: If $X$ is a CW complex then: a) $H_{k}(X^{n},X^{n-1})$ is zero for $k\neq n$ and is free abelian for $k=n$ with a basis in one-to-one correspondence with the n-cell of $X$. b) $H_{k}(X^{n})=0$...
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49 views

If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ [duplicate]

If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ I'm trying to prove the above which is a statement in Introduction to ...
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2answers
89 views

Give a cell decomposition of $[0, 1] \subseteq \mathbb{R}$

I'm trying to come up with examples to better understand the definition of the cell decomposition of a topological space $X$. The simplest example I could think of would be $X =[0, 1] \subseteq \...
3
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1answer
92 views

H-space multiplication for a CW-complex

Let $(X,\mu,e)$ be an $H$-space (so $\mu:X\times X\to X$ is a continuous map and $e\in X$ is a base point such that $\mu(\cdot,e)\simeq id_X\simeq\mu(e,\cdot)$) and a CW-complex such that $e$ is a $0$-...
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1answer
119 views

Attaching $2$-dimensional cell to $D^2$ gives the space $S^2/(x\sim -x)$

I am studying Algebraic Topology, and right now I am going through cell-attachment, which I have a pretty hard time to grasp. An "example" they give in the book is: Example: Define $X$ to be the ...
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1answer
111 views

Bijection between homotopy clases preserving basepoints and not preserving

Let us denote $<(X,x_0), (Y,y_0)>$ the set of basepoint-preserving-homotopy classes of basepoint-preserving maps $X \to Y$; and $[X,Y]$ the set of homotopy classes of maps $X \to Y$. There is a ...
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1answer
91 views

Homotopy domination of a pseudo projective plane

Recall that the pseudo projective plane $\mathbb{P}_n =\mathbb{S}^1 \cup_f e^2$ is obtained by attaching a 2-cell $e^2$ to $\mathbb{S}^1$ via the map $f:\mathbb{S}^1 \longrightarrow \mathbb{S}^1$ of ...
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1answer
119 views

An isomorphism of based homotopy maps $[X,S^1]$ to homomorphisms of their fundamental group.

This is the last question on a topology 1 exercise sheet, so hints aswell as solutions are highly appreciated! We are given a based connected CW-complex $(X,x)$, and the map $$W:[X,S^1]_*\rightarrow ...
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1answer
199 views

CW complex which is not a polyhedron

I understand that a CW complex is a generalization of the concepts of simplicial complex, and it is simple to visualize how any polyhedron can be seen as a CW complex, but since they're different ...
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1answer
105 views

Exercise Rotman 8.33 Algebraic Topology CW complex structure

If $(X,E)$ is a CW complex then so is $X \times I$. I have constructed the characteristic maps: We could give a cell decomposition of $X \times I$ by $E'' := \{ e \times a^0, e \times b^0 , e ...
2
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1answer
186 views

A subcomplex of an absolute CW-complex makes for a relative CW-complex

Let $X$ be CW-complex and let $Y$ be a subcomplex. I have to show that $(X,Y)$ is a relative CW-complex. I thought this would follow quickly, but I got stuck. The case when $Y\neq X_i$ for all $i\ge-1$...
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0answers
137 views

Fundamental Group of a Given Polygon

I have the following question that I'm trying to reason out. Here it is: Fix $n\in\mathbb{N}$ and let $V\subset\mathbb{R}^{2}$ be a closed polygon of $2n$ sides such that $\partial V=a_{1}\cup a_{...
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1answer
401 views

Finite and finite-dimensional CW complex

A finite CW complex is one with a finite number of cells while a finite dimensional CW-complex is one with no cells of dimension greater than a nonnegative integer $n$, we say in this case that $X$ is ...
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351 views

Show that a CW-complex is contractible without Whitehead theorem

I was trying to solve the following problem: Let $X$ a CW- complex such that the inclusion $X_{n-1} \hookrightarrow X_n $ are null-homotopic, then $X$ is contractible. I already know that Whitehead ...
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1answer
289 views

Is an image of a cellular map CW-complex?

Let $X$ and $Y$ be two CW-complexes and $f:X\longrightarrow Y$ be a cellular map. Is $f(X)$ a CW-complex? If the answer is no, then under what conditions $f(X)$ is a CW-complex? Thanks in ...
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1answer
23 views

Why can these two functions be composed in this diagram? - CW Complex

This is taken from Rotman's Algebraic Topology: In the red box in the image above, how are $\bar f \Phi_e$ and $\tilde G(\Phi_e \times 1)$ defined in terms of domain and codomain? By definition, $\...
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1answer
137 views

John Lee, CW construction, p139, components of nth skeleton

This is from John M Lee's Topological manifold and the CW construction theorem (p139). Theorem: Suppose $X_0 \subseteq X_1 \subseteq \cdots $ is a sequence of topological space satisfying the ...
2
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1answer
92 views

Category of a space bounded by length of stratification

I am reading May's 'More Concise Algebraic Topology' and I ran into something I couldn't figure out. Let $(X,*)$ be a based CW-complex. The category of $X$ is said to be less than $k$ if the ...
2
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1answer
181 views

CW-complex inclusion and pinch maps

In general a periodic map on a finite CW-complex leads to a periodic family of elements in $\pi^S_*$, although the procedure is not always as simple as in the above examples. Each of them has the ...
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1answer
38 views

Compactness of CW space continuous maps from it

The first problem is to prove that CW space is compact if and only if it is finite. It's pretty obvious to show that finite CW space is compact because finite collection of cells is compact and so ...
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1answer
388 views

Cell Structure of Three holed torus

How do I construct a torus as a cell structure? Visually I do not think I quite see the construction. Furthermore, how do I construct a 3-holed torus, that is, a 3 genus surface as a cell structure?
2
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1answer
69 views

lifting weak equivalences to cellular approximations

Lets say I have a weak homotopy equivalence $Y \rightarrow Y'$ between arbitrary topological spaces. Does the dashed arrow in $$\begin{array}{ccc} Y_{CW}&\dashrightarrow&Y'_{CW}\\\ \...
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2answers
1k views

Proving that a subcomplex of a CW Complex is also a CW Complex

The following proof is taken from Introduction to Topological Manifolds by John Lee I believe there is a (slight) error in the proof, where the author says "it is the disjoint union of its cells". ...
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1answer
104 views

Is $\mathbb{R}^\infty$ a CW complex?

I know that $\mathbb{R}^n$ has the structure of CW complex by considering lattices. What about $\mathbb{R}^\infty$?
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1answer
97 views

If $X\to X/G$ is a universal covering map of CW-complexes, then is $G$ always free on the cells of $X$?

Let $X\to X/G$ be a universal covering map of CW-complexes, then does $G$ always act freely on the cells of $X$? I know if the action of $G$ is free, then $X\to X/G$ is a normal covering map, I wonder ...
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0answers
58 views

Approximation by finite complex

Let $X$ be a simply connected space such that $H^k(X)$ is finitely generated for every $k$ and $H^k(X)=0$ for $k>n$. How can I find a finite $n$-(CW) complex $K$ homotopy equivalent to $X$? The ...
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0answers
154 views

boundary map of a CW complex

I am confused about boundary maps of a CW complexes. For example, let $X$ be $\mathbb{C}\times \mathbb{R}$. $t_1,t_2,\alpha$ are the transformations on $X$ such that $$t_1:(z,x)\mapsto (z+1,x)$$ $$...
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1answer
46 views

If $X_{\lambda}$ is a family of CW complexes, then $\tilde H_*(\lor_{\lambda}X_{\lambda}) \approx \sum \tilde H_*(X_{\lambda})$

If $\{X_{\lambda}: \lambda \in \Gamma\}$ is a family of CW complexes with basepoint, then $\tilde H_*(\lor_{\lambda}X_{\lambda}) \approx \sum \tilde H_*(X_{\lambda})$ $(\lor_{\lambda}$ means wedge) I'...
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1answer
322 views

If $X$ is a compact CW complex with CW subcomplex $Y$, then $H_k(X,Y)$ is finitely generated for every $k \ge 0$.

If $(X, E)$ is a compact CW complex with CW subcomplex $(Y, E')$, then $H_k(X,Y)$ is finitely generated for every $k \ge 0$. I see that to show $H_k(X,Y)$ is finitely generated I could also show that ...
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3answers
463 views

If $X$ is a CW complex, then the path components of $X$ are the components of $X$.

I'm self-learning Algebraic Topology from Rotman's Introduction to Algebraic Topology and I've come across this problem: If $X$ is a CW complex, then the path components of $X$ are the components ...
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1answer
247 views

If $(X,E)$ is a CW complex, then a subset $A$ of $X$ is closed iff $A \cap X ' $ is closed in $X'$ for every finite CW subcomplex $X'$ in $X$.

Prove: If $(X,E)$ is a CW complex, then a subset $A$ of $X$ is closed iff $A \cap X ' $ is closed in $X'$ for every finite CW subcomplex $X'$ in $X$. The $\Rightarrow$ proof in Rotman's Algebraic ...
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1answer
37 views

If $\Phi_e^{-1}(C \cap \bar e)$ is compact, then $C \cap \bar e$ is compact?

Could someone explain why what's in the red box is true? How does the compactness of $\Phi_e^{-1}(C \cap \bar e)$ imply the compactness of $C \cap \bar e$?
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1answer
166 views

In what sense are trivalent graphs “generic” 1-dimensional CW-complexes?

The Wikipedia article CW complex mentions that "trivalent graphs can be considered as generic 1-dimensional CW complexes", offering this explanation: Specifically, if $X$ is a 1-dimensional CW ...
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1answer
85 views

Transfinite induction on inductively constructed CW complex

Let $X$ be a 2-dimensional path-connected CW-complex and let $W \subset X$ be a subcomplex. Further let $I$ be an index set of the path-components of $W$. Consider the following iterative procedure: ...
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0answers
91 views

Working out a proof of how to calculate $H_k(\mathbb{R}\mathrm{P}^n)$

The theoretical setting Assume $ X_{-1}:=\emptyset, X_0 \subset X_1 \subset \dots \subset X\;$ is a CW-complex with pushouts $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-...
5
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1answer
673 views

Euler characteristic for CW complexes

Can someone help me of how to prove these two basic properties of Euler characteristics, but regarding finite $CW$ complexes. $a)$ If $A$ and $B$ are two subcomplexes of a finite $CW$ complex $X$, ...
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1answer
53 views

Understanding a property of regular CW-complexes

I'm studying CW-complexes and regular CW-complexes be a special case . It has a basic property as you can see in lemma $5.5$ in the picture above . I assume you are familiar with CW-complexes . Could ...
3
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2answers
565 views

Homology of direct limit/ homology of a CW complex

I have come upon a gap in lecture notes I am working through and would appreciate some help. I'm considering the following: Let $H_\ast$ denote singular homology. Given a filtration $$ \emptyset =: ...

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