# 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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### Understanding CW complex construction

Hatcher's construction of a CW complex is as follows (see page $5$ of Hatcher's Algebraic Topology): (1) Start with a discrete set $X^0$ , whose points are regarded as $0$ cells. (2) Inductively, form ...
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### Relative cell complexes in the undercategory $B/\mathscr{M}$ are relative cell complexes in $\mathscr{M}$ - why must we also assume $B$ is a complex?

$\newcommand{\M}{\mathscr{M}}\newcommand{\I}{\mathcal{I}}\newcommand{\J}{\mathcal{J}}$The context: the following is claimed in J. May's "more concise algebraic topology". We have some model ...
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### Proving Invariance of CW Complex Dimension

John Lee's Introduction to Topological Manifolds defines the dimension of a CW complex $X$ as the largest dimension of a cell in $X$. The author claims "The fact that this well defined depends on ...
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### CW complex for Möbius strip and its homeomorfisams

I have to find CW complex for Mobius Strip. I can find a CW complex for a Möbius strip with more cells (one 0-cells, two 1-cells and a single 2-cell). My cells are points $A=(0,0) \approx (1,1)$, part ...
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### Can we glue CW complexes to themselves? [closed]

My understanding: CW complexes are constructed by inductively attaching $n+1$ cells to an $n$-skeleton where a 0-skeleton consists of a discrete set of 0-cells (points). What I fail to understand: ...
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### proving that the inclusions $i:X \to CX$, $j:Y \to C_f$, $j':Y \to M_f$ have the homotopy extension property

Let $X$ be a topological space, and $f:X \to Y$ a continuous map. I want to show that the inclusions $i: X \to CX, \: j:Y \to C_f, \: j':Y \to M_f$ all have the homotopy extension property. Where $CX$...
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### For a cw-complex the pair $\left(X,X^n \right)$ is n-connected

Hello I am currently self-learning about (algebraic-)topology using Hatchers book. My definition of $n$-connected is that $\pi_k\left(X,X^n\right)=0$ for all $1\le k \le n$.I do know about the ...
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### higher-degree attaching maps in cell complexes

I'm new to algebraic topology -- sorry if this is a silly question or covered in some obvious place. I didn't find it in Hatcher or other questions on this site. What I get so far: Consider an $n$-...
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### In cellular homology, how should we define the degree of a map between $0$-spheres?

Cellular homology is one of many homology theories available for 'nice' spaces. There are doubtless many equivalent definitions for it, in increasing levels of abstraction, but I want to ask about the ...
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### CW-complex of Möbius band without 0-cells

So proofwiki, and some other sites, claim the Euler-characteristic of the Möbius strip is 0-1+1=0. Relying on the fact that a Möbius strip has no vertices, i.e. 0-cells. However I can nowhere find a ...
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### Cohomology class $H^k(X; \pi_{k-1}(Y,y_0))$ as obstruction

I am following these notes http://scgp.stonybrook.edu/wp-content/uploads/2018/09/lecture-1.pdf on obstruction theory. Let $X$ be a CW complex and denote by $X^{(k)}$ its $k$-skeleton. Suppose that $Y$ ...
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### Does gluing polyhedra together give you a CW complex?

If I take a finite collection of polyhedra and glue their faces together (potentially gluing two faces of the same polyhedron together) to get a closed 3-manifold, is this always a CW-complex? Can I ...
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### Degree of maps of product CW complex

If we treat the points of $S^3$ as quaternions i.e. $S^3=\{y_0+y_1i+y_2j+y_3k \mid y_0^2+y_1^2+y_2^2+y_3^2=1\}$, let $X=\mathbb{R}P^3$ by identifying antipodal points of $S^3$. Then quaternion ...
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Let me first state my definition of cell complexes. Definition 1: Cellular decomposition of a topological space $X$ is a cover $\mathcal{E}$ of space $X$, equipped with a function deg: \$\mathcal{E} \...