Questions tagged [cw-complexes]

For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

Let $B^n$ denote the $n$-dimensional closed ball.

If $X$ is a topological space and $\varphi : \partial B^n \to X$ is a continuous map, the adjunction space is $X\cup_{\varphi} B^n := (X\coprod B^n)/\sim$ where $\sim$ identifies $x$ with $f(x)$. The process of going from $X$ to $X\cup_{\varphi} B^n$ is often referred to as attaching an $n$-cell and the map $\varphi$ is called the attaching map.

Let $X_0$ be a discrete space. Let $X_n$ be a space which can be obtained from $X_{n-1}$ by attaching $n$-cells. Then the space $X = \bigcup_{n=0}^{\infty}X_n$, topologised appropriately, is called a CW complex, and the spaces $\bigcup_{n=0}^kX_n$ are called the $k$-skeletons of $X$.

Surprisingly, not every topological space is a CW complex. For example, the Hawaiian earring does not even have the homotopy type of a CW complex.

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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 of simplicial complexes (without the requirement…
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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...
LASV
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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. The proof contains some mistakes I feel (perhaps…
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CW complex structure of the projective space $\mathbb{RP}^n$

I'm trying to understand the CW complex structure of the projective space $\mathbb{RP}^n$, but some things are unclear. I understand we start by identifying $\mathbb{RP}^n$ with $S^n/R$ where $R$ is the equivalence relation identifying antipodal…
Tom
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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 read well the answer, it starts by stating that it…
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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 the homotopy type of $X$, not on its decomposition as a…
Potato
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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 consisting of points that have a representative in…
Ben Blum-Smith
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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 (at least this is my intuition about this) a…
Tobias Kildetoft
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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 silly examples like $X=pt$ , $Y$ some…
KotelKanim
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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.
Aspirin
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What manifolds $M$ have a $CW-$structure so that the $n-$skeleton, $M_n$, is a manifold for all $n$ aswell?

If you have a $CW-$structure on a connected manifold $M$ we obtain a filtration $M_n$ of $M$ where $M_n$ is the $n-$skeleton. If in addition we have that the $M_n$ are also all manifolds (like with the standard $CW-$decomposition of $\mathbb RP^n$…
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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 the higher homotopy groups each be a specified…
James Cameron
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Covering image of a connected CW-complex need not be a CW-complex.

Problem: Let $X$ be a connected CW-complex, and $Y$ be a connected topological space. Suppose $p: X\to Y$ is a covering map. Does there exist a CW-structure on $Y$? More generally, is $Y$ homotopically equivalent to a CW-complex? Note that the…
Sumanta
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Whitehead product and a homotopy group of a wedge sum

Note : this question has been crossposted on the mathematics Overflow. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove the following isomorphism : $$\pi_{n+k+1}(X\vee…
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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 the degree of the map $S^{n-1}_\alpha \rightarrow…
Bill
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