# 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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### "Introduction to Topological Manifolds" John Lee, Theorem 10.15

How do I know such a point $v$ below is guaranteed to exist? Wouldn't 3 closed disks side by side of radius 1, centered at (0,0), (2,0), and (4,0) in $\mathbb{R}^2$ be a connected finite CW complex? ...
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### Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
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### Is $X\times \{0,1\}$ in the unreduced suspension contractible?

I am trying to prove that the unreduced suspension of a CW complex $X$ is a CW complex and I am using the idea of the relative CW complexes given here Suspension of a CW complex my idea (in terms of ...
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### Fundamental group of the complementary of a cell complex

I am trying to understand the proof page 23 of this document which provides a method for computing fundamental group of the complement of a cell complex. Im doing an internship and trying to ...
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### Partitioning Cellular Prism Operator?

In singular homology, we have the prism operator $P : C_{n}(X) \rightarrow C_{n+1}(Y)$ between singular chains, and for a (singular) $n$-simplex $\sigma$, $P(\sigma)$ decomposes into $(n+1)$ simplices ...
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### Prove that removing an open 2-cell from $S^2$ results in a contractible space

Let $X$ be a cellular decomposition of $S^2$. I want to show that if $r\in X^{(2)}$ then $X\setminus \text{Int}(r)$ is a contractible space. I don't know much topology so I don't know if this it ...
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### Pushfoward of a CW complex structure by a covering map

Let $p:Y\to X$ be a covering map. If $X$ has a CW complex structure, then we can give a CW complex structure on $Y$ so that $p$ becomes a cellular map, by lifting the characteristic maps (cf. Euler ...
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