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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51
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1answer
9k views

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 ...
34
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2answers
829 views

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. ...
29
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2answers
6k views

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...
20
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1answer
1k views

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 ...
17
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1answer
314 views

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 ...
16
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1answer
497 views

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 (...
16
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2answers
493 views

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 ...
15
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0answers
2k views

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 ...
12
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1answer
6k views

Cartesian product of two CW-complexes

Let's $A$ and $B$ are CW-complexes. How to construct CW-complex $A\times B$? Thanks.
12
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3answers
1k views

Intuition behind CW complexes

These things are the bane of my existence in mathematics. I feel that I can't find any clear examples of these things anywhere. This is a vague question, but how exactly do we intuitively visualize ...
11
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2answers
1k views

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 ...
11
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2answers
901 views

Factorization of a map between CW complexes

I've been working on problem 4.1.16 of Hatcher's Algebraic Topology and am at a complete impasse. The problem is as follows: Show that a map $f:X→Y$ between connected CW complexes factors as a ...
11
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3answers
2k views

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 ...
9
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2answers
2k views

Is a CW complex, homeomorphic to a regular CW complex?

Is a CW complex, homeomorphic to a regular CW complex ? Regular means that the attaching maps are homeomorphisms (1-1). In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open (...
9
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1answer
1k views

Morphism induced by a cellular map between CW-complexes

I'm trying to understand cellular homology as a functor from the category (CW-complexes, cellular maps) to the category of abelian groups sequences. Let $X,Y$ be fixed CW-complexes. My lecturer ...
8
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2answers
950 views

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 ...
8
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1answer
1k views

The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?
8
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0answers
133 views

Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
8
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0answers
64 views

CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
8
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0answers
194 views

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 $f:K_1\rightarrow K_2, g:K_2\rightarrow K_3$,...
7
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2answers
1k views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
7
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2answers
457 views

Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
7
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1answer
204 views

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 ...
7
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1answer
630 views

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 ...
7
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1answer
2k views

Fundamental group of a CW complex only depends on its $2$-skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1(X)$ only depends on the $2$-skeleton of $X$ for any CW complex $X$. I am very ...
7
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1answer
228 views

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 ...
7
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0answers
142 views

$Mat_{ \infty,n }(\mathbb C)$ 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 ...
7
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0answers
238 views

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,...
7
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0answers
117 views

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 $CW$-...
6
votes
2answers
875 views

A map which is trivial on homology but not on cohomology?

Is there a map $f:X\to Y$ of connected CW-complexes which induces the trivial map $f_*=0:H_i(X,\Bbb Z) \to H_i(Y,\Bbb Z)$ for all $i\ge 1$, but with the property that the induced map on cohomology $f^*...
6
votes
1answer
3k views

A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: If ...
6
votes
2answers
208 views

Specific examples of Eilenberg-Maclane spaces?

Given an integer $n$ and a group $G$ (abelian if $n \geq 2$), it's always possible to construct a $K(G,n)$ as a cell complex. The standard procedure is to choose a presentation $\langle S | R \rangle$ ...
6
votes
1answer
1k views

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 ...
6
votes
3answers
567 views

existence of CW complex construction

It looks like CW complexes come up a lot when studying spaces in algebraic topology. However, something that doesn't seem to be mentioned a lot is whether a given topological space as a CW complex. Is ...
6
votes
2answers
3k views

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 $\Phi_\sigma:\mathbb D^n \to X$ ...
6
votes
2answers
279 views

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 $X=\bigcup X^n$, ...
6
votes
3answers
292 views

Decoupling the algebra from the topology in cellular homology

The story of singular homology is pretty straightforward. One starts by constructing the singular chain complex functor $S : \mathbf{Top} \to \mathbf{Cha}$ (category of chain complexes with chain maps)...
6
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1answer
337 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
6
votes
1answer
738 views

Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
6
votes
1answer
487 views

What exactly is the CW complex structure on a geometric realisation?

This is likely a silly question. Definitions: $\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$ $\bullet$ Given $f: \underline{m} \to \underline{n}$ in $\...
6
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1answer
125 views

Hatcher exercise 3.2.14 - on the surjectivity of the map $q: \mathbb{R}P^2 \to \mathbb{C}P^1$ in cohomology $H^2$

I'm working through some exercises from Hatcher and I'm having trouble understanding the first part of exercise 3.2.14, which goes as follows: Let $q: \mathbb{R}P^{\infty} \to \mathbb{C}P^{\infty}$ ...
6
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0answers
64 views

Intuition for Freudenthal Suspension

One version of the Freudenthal suspension theorem is the following: Suppose a CW complex $X$ is a union of two subcomplexes $A,B$ with $A\cap B\neq\emptyset$ connected and nonempty. If $(A,A\cap B)$...
5
votes
2answers
507 views

An infinite dimensional CW complex always has infinitely many non-trivial homology groups?

Let $X$ denote an infinite dimensional CW complex, I wonder if $H_n(X)\ne 0$ for infinitely many $n$'s. I think we may need to use cellular homology. The only thing I have got is that since $X$ is ...
5
votes
3answers
312 views

Wedge sum of spheres is the quotient $X^n/X^{n-1}$

As in the title, I want to prove that $\bigvee_jS_j^n=X^n/X^{n-1};\ X$ is a $CW$ complex and $X^n$ and $X^{n-1}$ are the $n-$ and $n-1$-skeleta. Below, I present a sketch of an attempt using pushouts, ...
5
votes
2answers
182 views

Is a sequential limit of homeomorphisms a homeomorphism?

Suppose that we have a diagram in the category Top of the following form $$ \cdots X_2 \to X_1 \to X_0=X $$ where the arrows are homeomorphisms. Is it true that that the natural morphism $lim_i X_i \...
5
votes
1answer
473 views

CW complex structure of geometric realization

In Ralph Cohen's notes on the topology of fiber bundles he makes the following claims: on pp.69, he says the geometric realization of a simplicial set is a CW complex on pp.70, he says the geometric ...
5
votes
1answer
768 views

Associativity of the smash product on compactly generated spaces

Given pointed topological spaces $X$ and $Y$, their smash product is the space $$ X \land Y = \frac{X \times Y}{X \times \{ y_0\} \cup \{x_0\} \times Y}, $$ where $x_0$ and $y_0$ are the distinguished ...
5
votes
1answer
1k views

Moore Spaces: explicit CW-complex for $M(\mathbb{Z}_m, n)$

Given an abelian group $G$ and an integer $n \ge 1$ we can construct a $CW$ complex such that $H_n(X) \cong G$ and $\tilde{H}_i(X)=0$ for all $i \neq n$. We call this $CW$ complex a Moore space and ...
5
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1answer
780 views

Is every finite CW complex is homotopic to simplicial complex?

Is every finite CW complex is homotopy equivalent to a simplicial complex?
5
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1answer
1k views

Some questions about cellular homology and cohomology

Consider the CW structure on $\mathbb{RP}^n$ given by one cell in every dimension. This gives rise to the cellular complex $C_\bullet(\mathbb{RP}^n)$ which is generated by a single element $c_i$ for ...

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