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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Homotopy type of manifolds homeomorphic to the interior of a compact manifold with boundary

Let $M$ be a topological manifold which is homeomorphic to the interior of a compact manifold with boundary. This extends the class of compact manifolds and is sometimes called being of finite topological type. What can be said about the CW-complex…
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Use van Kampens theorem to compute the fundamental group of a torus with a ball attached via a map

Use van Kampens theorem to compute the fundamental group of the following space: $A$ is a torus with an open disk $D$ removed. Let $f:\partial B \rightarrow \partial A$ be a map from the boundary of a 2-ball $B$ to the boundary of $A$ winding twice…
TuoTuo
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CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\limits_{i\in \mathbb{N}} X_i$ of finite cell…
user197416
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Isomorphism of Homotopy groups across a filtration

Let $X$ and $Y$ be CW complexes. Let $sk_{\bullet}(X)$ and $sk_{\bullet}(Y)$ denote the canonical skeleta filtrations of $X$ and $Y$, respectively. Suppose that we have isomorphisms on homotopy groups $\pi_n(sk_{i}(X))\simeq\pi_n(sk_{i}(Y))$ for all…
user84563
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Integral homology groups of the complexe projective n-plane

I am reading "Morse theory" by Milnor and on page 27 we have proved that the homotopy type of CP^n is of a CW-complex of the form : a 0-cell attached to a 2-cell attached to a 4-cell ... attached to a 2n-cell.µ Then Milnor says it follows that the…
Amandine
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Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex structure, then it is a manifold? Furthermore, to prove that…
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Questions on CW complex structure

REMARK: I had already posted these questions, about one hour ago, but one of the questions was not what I meant. I am in the beginning of my studies in Algebraic Topology and am studying CW complexes and identification spaces. I have two questions…
user194469
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Characteristic function is an identification function

Every characteristic function $\Phi_{\beta}^b: E^n_{\beta} \to e^{-n}_{\beta}$ is an identification function. My book says the following: This follows from the fact the the CW complex $X$ has $X^n$ a quotient space of $X^{n-1} \sqcup…
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Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the corresponding 1-skeleton, give a presentation of…
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A Small Fact About CW Complexes (Proof Checking)

Let $X$ be a topological space and $\mathcal E$ and $\mathcal E'$ be two finite CW complex structures on $X$. Let $e\in \mathcal E$ and $e'\in \mathcal E'$ be two top dimensional cells such that $e\cap e'\neq \emptyset$. Assume further that $e'$ is…
caffeinemachine
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is this map an homotopic equivalence of pairs from (disk, sphere) to the disk seen attached to a space?

Hello I was studying https://www2.warwick.ac.uk/fac/sci/maths/people/staff/vincent/cohomology.pdf On page 18 given a space $A$ and a map $f:\mathbb{S}^{n-1} \rightarrow A$ he defines the cone $X := A \cup_f D^n$ I was trying to show that $(D^n,…
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CW approximation in Hatcher

On page 352 of Hatcher's online text located here: https://www.math.cornell.edu/~hatcher/AT/ATpage.html , with the relevant chapter here: https://www.math.cornell.edu/~hatcher/AT/ATch4.pdf , he proves the CW approximation theorem. He does this by…
qwert4321
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wedge sum of $S^1$ is a finite CW complex

I'm in trouble with CW complexes. I want to know, how to prove that the wedge sum $S^1\vee S^1$ of $(S^1,x_0)$ and $(S^1,y_0)$ is a finite CW complex, $y_0$ and $x_0$ are base points My explanation is that you have the two 0-cells $x_0,y_0$ and two…
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Getting intuition for Munkres' 2 dimensional CW complex construction:

I've been working my way through Munkres's Topology, and came across the following question which I'm having some difficulty wrapping my head around. Prove the following: Theorem: If G is a finitely presented group, then there is a compact…
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What is the name of the $(k-1)$ faces of a $k$ cell?

Is there an own name of the $(k-1)$ cells that are attached to a given $k$ cell (or in other words: of the $(k-1)$ cells that intersects a given closed $k$ cell, or yet another words: of the $(k-1)$ cells that are in the boundary of a given closed…
mma
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