# 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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### Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
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### Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
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### CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
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### Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow K_3$,...
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### $Mat_{ \infty,n }(\mathbb C)$ of max rank is contractible

We want to show that for a given $n\in\mathbb{N}$, the set of all the $\infty\times n$ matrices $Mat_{\infty,n}(\mathbb{C})$ of max rank $n$ form a contractible space. This set is obtained by taking ...
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### Eilenberg-Zilber Theorem Proof

Let $X, Y$ be CW complexes and the following proof (from J. P. May's "A Concise Course in Algebraic Topology"; see page 102. Here the full pdf document: https://www.maths.ed.ac.uk/~v1ranick/papers/...
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### Restriction of Attatching Map for CW Complexes Closed

A CW complex $X$ is a Hausdorff space and can be interpreted as a colimit $X = colim_k X_k$ of cells $X_k$ which satisfy can inclusion relations $X_{k-1} \subset X_k$ for each $k$ and are defined ...
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### The space obtained by attaching a subspace of a topological space to itself.

Just asking a special case of attaching, it seems pretty vague to me. Let $Y$ be a space and $X$ be its subspace. Then the space obtained by $Y$ attaching its subspace $X$ via $\iota$ (the embedding) ...
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### Weak product of Eilemberg MacLane spaces

I'm studying some homotopy theory of topological monoids from the book Algebraic Topology from a Homotopical Viewpoint. I'm trying to understand the corollary below. I'm stuck on the first claim of ...
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### Using Homotopy Extension Propperty to prove that $S^{\infty}$ is contractible

I know that the question: ''Is $S^{\infty}$ contractible?'' has already been asked, but none of those post seem to answer it the way I need (Maybe they do, and my lack of knowledge prevents me from ...
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### 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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### 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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### 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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### 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}{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-...
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### $G$ acts freely and cocompactly on $Y$ by permuting the cells, then $G$ is a factor group of $\pi_1(Y/G)$?

Let $Y$ be a connected CW-complex and $G$ a group acting freely on $Y$ by permuting the cells. We assume the action on $Y$ to be cocompact so that $X = Y/G$ is a finite CW-complex. Then how to see $G$ ...
Let $X$ be a $CW$-complex and let $G$ be a group acting "cellularly" on $X$, that is, for each $g\in G$, the induced homeomorphism $\phi_g:X\to X$ takes cells to cells. Also assume that the action is ...
Consider pairs $(X,A),(Y,B)$ of $CW$-complexes, i.e. $X$ is a CW-complex and $A\subseteq X$ a CW-subcomplex and the same for $(Y,B)$. I want to know the following: How to prove in singular (...