# 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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### Simplicial Complex vs Delta Complex vs CW Complex

I am a little confused about what exactly are the difference(s) between simplicial complex, $\Delta$-complex, and CW Complex. What I roughly understand is that $\Delta$-complexes are generalisation ...
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### Paracompactness of CW complexes (rather long)

I finished reading Lee's 'introduction to topological manifolds' (2nd edition) and I'm currently tying up some loose ends. One thing I can't understand is the proof of paracompactness of CW complexes. ...
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### CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
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### Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
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### Cell decomposition for $\mathbb{C}P^n$ that has $\mathbb{R}P^n$ as a subcomplex?

Real projective space $\mathbb{R}P^n$ embeds in a natural way in complex projective space $\mathbb{C}P^n$. (Using standard projective coordinates on $\mathbb{C}P^n$, $\mathbb{R}P^n$ is the subspace ...
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### Can torsion in the fundamental group happen in “the real world”

Suppose that $X$ is a CW-complex such that $\pi_1(X)$ has non-trivial torsion. Does this imply that $X$ cannot be embedded in $\mathbb{R}^3$? Intuitively, this seems like it should be clear, since (...
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### Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
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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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### Cartesian product of two CW-complexes

Let's $A$ and $B$ are CW-complexes. How to construct CW-complex $A\times B$? Thanks.
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### Intuition behind CW complexes

These things are the bane of my existence in mathematics. I feel that I can't find any clear examples of these things anywhere. This is a vague question, but how exactly do we intuitively visualize ...
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### Constructing $\pi_1$ actions on higher homotopy groups.

I am working on exercise 4.2.7 of Hatcher, which is to construct a CW complex $X$ with arbitrary homotopy groups and a prescribed action of the fundamental group on these homotopy groups (so making ...
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### Factorization of a map between CW complexes

I've been working on problem 4.1.16 of Hatcher's Algebraic Topology and am at a complete impasse. The problem is as follows: Show that a map $f:X→Y$ between connected CW complexes factors as a ...
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### Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta}$ where $d_{\alpha\beta}$ is ...
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### Is a CW complex, homeomorphic to a regular CW complex?

Is a CW complex, homeomorphic to a regular CW complex ? Regular means that the attaching maps are homeomorphisms (1-1). In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open (...
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### Morphism induced by a cellular map between CW-complexes

I'm trying to understand cellular homology as a functor from the category (CW-complexes, cellular maps) to the category of abelian groups sequences. Let $X,Y$ be fixed CW-complexes. My lecturer ...
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### Visualizing products of $CW$ complexes

I'm learning about products of CW complexes. The sources I've seen talk about the matter as follows: given topological spaces $X$ and $Y$ with a given CW decomposition, we can then form a CW ...
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### The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?
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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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### cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
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### Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
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### Is every space homology equivalent to an Eilenberg–MacLane space?

Homology equivalence may be defined as follows (any other ways could be not equivalent to one below): $X \sim Y$ if there exist two map $f : Y \to X$, $g : X \to Y$, such that $(fg)^* = id_{H(X)}$ and ...
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### Finding the degrees of the attaching map of the $2$-cell of the torus

I am trying to calculate the degrees of the attaching map of the two cell of the torus. I have the following cell structure: The $2$-cell is $e_2$ and the $0$-cell (all four corners) is $e_0$. I ...
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### Fundamental group of a CW complex only depends on its $2$-skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1(X)$ only depends on the $2$-skeleton of $X$ for any CW complex $X$. I am very ...
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### Computing homology of square with all vertices identified.

I'm trying to compute the homology of $X = (I \times I)/\sim$, where $(0,0)\sim (0,1) \sim (1,0) \sim (1,1)$. I want to do this via cellular homology, using degrees, etc, but I don't got that very ...
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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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### Moore Spaces: explicit CW-complex for $M(\mathbb{Z}_m, n)$

Given an abelian group $G$ and an integer $n \ge 1$ we can construct a $CW$ complex such that $H_n(X) \cong G$ and $\tilde{H}_i(X)=0$ for all $i \neq n$. We call this $CW$ complex a Moore space and ...
Consider the CW structure on $\mathbb{RP}^n$ given by one cell in every dimension. This gives rise to the cellular complex $C_\bullet(\mathbb{RP}^n)$ which is generated by a single element $c_i$ for ...