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.

719 questions
8
votes
1 answer

Showing that the dimension of a CW complex is well defined.

This is from page 204 of Rotman's An Introduction to Algebraic Topology. After some elementary definitions and facts about CW copmlexes, exercise 8.27 asks: Define the dimension of a CW complex $(X,E)$ to be $$\mbox{dim…
Dilemian
  • 1,101
  • 10
  • 29
8
votes
2 answers

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 decomposition of $X \times Y$ by understanding the…
Eric Auld
  • 26,353
  • 9
  • 65
  • 174
8
votes
1 answer

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 called the attaching map of $e_2$ $f_2$ and defined…
a student
  • 4,125
  • 19
  • 47
8
votes
0 answers

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 $f:K_1\rightarrow K_2, g:K_2\rightarrow K_3$, such that the induced homomorphisms $f_*, g_*$…
Samarkand
  • 461
  • 2
  • 13
7
votes
2 answers

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, plus some other requirements. They are endowed…
7
votes
3 answers

Characteristic map of a n-cell in a CW complex

I have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to X$ be the characteristic map. Is there any relation…
user83496
  • 363
  • 2
  • 10
7
votes
1 answer

$\pi_1$-equivalence of CW-complexes

Definition. We call $\pi_1$-equivalence the closure of a binary relation between topological spaces "there is a continuous mapping inducing an isomorphism of fundamental groups" to an equivalence relation. Questions: Are there two…
7
votes
1 answer

Any two non-separating curves on a surface are equivalent

Problem: Let $\Sigma$ be a orientable surface possibly non-compact with boundary. Let $C,$ and $D$ be two simple-closed curves(smooth embedding) on $\Sigma\backslash \partial \Sigma$ such that both $\Sigma\backslash C$, and $\Sigma\backslash D$…
7
votes
2 answers

Is a closed embedding of CW-complexes a cofibration?

It is a standard fact that the inclusion of a sub-CW-complex into a CW-complex is a cofibration, it follows from the fact that the inclusions $S^k\to D^{k+1}$ are, and that they are preserved by pushouts. My question is about a general closed…
7
votes
0 answers

$Mat_{ \infty,n }(\mathbb C)$ 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 the limit on $k \geq n$ in $Mat_{k,n}(\mathbb{C})$…
7
votes
0 answers

Associativity of join for CW-complexes

I am currently self-studying a course in algebraic topology and one of the problems I encountered is to prove that the join operation defined as $$X \ast Y=X\times Y\times I/(x_1,y,1)\sim(x_2,y,1), (x,y_1,0)\sim(x,y_2,0)$$ is associative for…
Sergey Guminov
  • 2,732
  • 1
  • 9
  • 16
7
votes
1 answer

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 $(gf)^* = id_{H(Y)}$ (where homology is taken…
Byobe
  • 533
  • 3
  • 10
7
votes
1 answer

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of spaces which are homotopy equivalent to finite CW…
7
votes
1 answer

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 well. It is clear to me that we start with one…
Ivo Terek
  • 71,235
  • 11
  • 84
  • 208
7
votes
0 answers

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even $CW$-complex) with $\pi_1(X)=\mathbb Z_2$? If it is…
pw1822
  • 688
  • 3
  • 14
1 2
3
47 48