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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Question 1 in the appendix on CW complexes in Hatcher's Algebraic Topology book.

I am asked to show that a covering space of a CW-complex is also a CW-complex with cells projecting homeomorphically onto cells. Do you have a reference for this question? Thanks in advance.
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Is a characteristic map in CW complex a quetient map?

Let $X$ be a CW complex and $\Phi : D \rightarrow \bar e$ be the characteristic map for an open cell $e$. I wonder whether $\Phi$ is a quotient map. I konw it is surjective. But I cannot prove that $\bar e$ has the quotient topology induced by…
Jeong
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Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a subcomplex of X. edit 0: As Lee Mosher pointed…
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cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring $H^*(\mathbb{R}P^{n+2}\setminus \mathbb{R}P^{n};\mathbb{Z}_2)$ and…
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CW -complex structure of boundary of a manifold

Given a CW-complex structure of manifold with boundary. Is there any natural way to construct CW-complex structure of its boundary? Thanking you.
Surojit
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On embedding a sort of $CW$ complexes to a Euclidean space.

I'd like to know if a finite dimensional, locally finite, $CW$ complex with countable cells can always be embedded to a Euclidean space. All I know is that it holds in the case $\dim=1$.
Censi LI
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CW-complex definition

I've the following definition a (finite) CW-complex of dimension $N$ is a topological space $X$ defined in the following way: $X^0$ is a discrete set of points $\forall 0
Louis
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Exercise 11, chapter 4, section 1 of 'Algebraic Topology' by Hatcher

I'm stuck in front of this exercise: Let $X$ be a CW-complex. If there exists an increasing sequence of sub-complexes $X_1\subset X_2\subset\cdots$ such that $X=\bigcup X_i$ and each $X_i$ is contractible in $X_{i+1}$, i.e. the inclusion $X_i\to…
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Reducing a path in a CW-complex

Let $X$ be a CW-complex with skeleta $X_0 \subset X_1 \subset \cdots $ Let $\gamma \colon [0,1] \rightarrow X$ be a path between points $x,y \in X_0$. Because $[0,1]$ is compact, Im$\gamma \subset X_n$ for some $n > 0$. (I think.) I want to…
Open Season
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Serre fibration and CW-complexes

Suppose that $p:X\rightarrow E$ is a Serre fibration. I know the definition: then $p$ has the right lifting property with respect to all inclusions $I^n\rightarrow I^{n}\times I$. Now it seems that one can equivalently say that one has the right…
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CW approximation

I was reading several proofs of the CW approximation theorem. If $X$ is a space then the idea is to make $n$-equivalences $f_n:K_n \to X$ where $K_n$ is a $n$-dimensional CW-complex. This goes by induction. Given $f_n$, the first part consists of…
user42761
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Obtain the Cayley Graph of $\mathbb Z\star \mathbb Z_2$.

I am trying to get the universal cover con $X=\mathbb RP^2\vee S^1$. Its fundamental group is $\mathbb Z\star \mathbb Z_2 =\langle a,b\mid b^2\rangle$ so it is reasonable to compute the Cayley Complex in order to get the universal cover of…
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Does always exist a CW-decomposition of topological space?

Does always exist a CW-decomposition of topological space ?
davis
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Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it is continuous? I am interested in a formal…
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