# 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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### Cellular Approximation for Homeomorphism on a Compact Surface.

In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows: Suppose $M$ is a ...
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### What are some uses of non-locally finite CW complexes?

A CW complex can be non-locally finite. An example of one is obtained by attaching a disk by its boundary to each point of $(0, 1)$, together with the end-vertices $\{0\}$ and $\{1\}$. When proving ...
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### Quotient of finite pointed CW complex is a CW complex

Let $X$ be a finite pointed CW complex (where the basepoint forms a $0$-cell) and $A$ be a CW subcomplex of $X$ containing its basepoint. I am trying to determine if $X/A$ is also a finite pointed CW ...
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### On abelian spaces

By Allen Hatcher's book, a space is called abelian if it has trivial action of $\pi_1$ on all homotopy groups $\pi_n$, since when $n=1$ this is the condition that $\pi_1$ be abelian. My quesion is ...
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### Two CW complexes with the same number of $n$-cells for each $n$ homeomorphic?

i am trying to understand Theorem 5.20 in Lee's Introduction to Topological manifolds I am interested in the uniqueness part. Does this mean that two CW complexes with the same number of $n$-cells ...
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### CW structure on the product $X\times Y$ of two CW-complexes $X$ and $Y$

I do understand that the $k$ -cells of $X\times Y$ are $\{e^i_X\times e^{k-i}_Y\}$ where $e^i_X$ is an $i$-cell of $X$ and $e^{k-i}_Y$ is a $k-i$ cell of $Y$ . I have the attaching maps from the CW-...
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### Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells?

Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells? Euler Characteristic of $\mathbb{R}P^2$ is $\chi(\mathbb{R}P^2)=1$ Since it has 1 0-Cell,1 1-Cell and 1 2-...
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### Reidemeister Torsion Example with Lens Spaces

In Andrew Ranicki's notes on Reidemeister torsion (https://www.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf), he gives the following example with lens spaces: where I copied the beginning of the ...
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### turn cofibration to fibre bundle

Suppose $X$ a CW complex, Y its subcomplex, $T$ is any parameter space. If we have map: $Y \to X \to X/Y$, the first map is inclusion and the second map is collapsing map, if we apply all spaces with ...
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### The dependence of the homotopy type on the attaching map

Suppose we have $\vee_{i=0}^k S^2$ the two skeleton and $h:S^3\to \vee_{i=0}^k S^2$ the attaching map to attach cell $e^4$. My question is, if two attaching map $h_1$ and $h_2$ are homotopy, can we ...
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### Is a topological space that is homotopy equivalent to a CW-complex locally path-connected?

While describing Quillen’s $+$-construction in his book Algebraic K-Theory, V. Srinivas assumes that his topological spaces are equivalent to CW-complexes and are path-connected. As universal ...
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I would like to compute $H_i(S^1\times S^n;\mathbb{Z})$ for $i\geq3$. I believe one approach is to use the Mayer-Vietoris sequence. If we take the admissible cover consisting of $S^1\times S^n\... 0 votes 1 answer 68 views ### Construct a 2-dimension CW-complex whose fundamental group is Z$\times Z/2

2-D CW-complex. If Z is a CW complex with one 0-cell. For simplicity, the $\pi_1(X)$ has presentation where the generators are the 1-cells and the relation come for the 2-cells. More precisely, each 1-...
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### Contractibility and uniform contractibility of metric spaces not homotopy equivalent to CW-complexes

I'm looking for an example of a metric space $X$ that is not contractible but satisfies the following "uniform contractability" condition: for every $r>0$ there exists $s\geq r$ such that ...
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### CW complex which is a union of contractible subcomplexes with contractible intersection is contractible

Im trying to show that if X is a CW complex, A,B $\subset$X are contractible sub complexes with contractible intersection and $X=A\cup B$, then X is contractible. We can say that X/A$\cap$B is ...
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### $X$ is a deformation retract of its mapping cylinder

I am trying to show the following: If $f : X \to Y$ is a homotopy equivalence and moreover a cellular map between finite CW-complexes, then $X$ is a deformation retract of $M_f$ via a cellular map. ...
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### CW complexes with cofibration morphisms

Let $CW$ be the category of CW complexes and $CW_{cof}$ be the wide subcategory whose objects are CW complexes and morphisms given by inclusions of subcomplexes (i.e. cofibrations in the model ...
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### Twisting Products in Hatcher's Algebraic Topology

In Allen Hatcher's book Algebraic Topology, in several places is used the terminology 'twisted product'; eg on page 338: Among other things, fibrations allow one to describe, in theory at least, how ...
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### Calculating the simplicial homology of a tetrahedron.

I want to calculate the simplicial homology of the $\Delta$-complex figure on the right: I believe that it consists of $2$ 0-simplices, four $1$-simplices and three $2$-simplices. So I am now in the ...
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### What are the morphisms in the 'category of CW-pairs'?

A CW-pair consists of a CW complex $X$ (with cell decomposition $\mathcal{E}\equiv\{e_\alpha\}_{\alpha\in I}$) together with one of its subcomplexes $A$ (a closed subspace of $X$ consisting of a union ...
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