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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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...
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.
4
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2answers
3k views

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology ring ...
5
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1answer
123 views

Assigning integer to finite CW complex such that following hold. [closed]

For each $n \in \mathbb{Z}$, is there a unique function $\varphi$ assigning an integer to each finite CW complex, such that the following hold? $\varphi(X) = \varphi(Y)$ if $X$ and $Y$ are ...
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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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 ...
1
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1answer
135 views

Confusion about topology on CW complex: weak or final?

The topology of the CW complex is defined to be the weak topology: given the sequence of inclusions of the skeleta $X_0 \subseteq X_1 \subseteq_ \cdots$ a subset $A \subseteq X = \cup X_i$ is open iff ...
6
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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 $\Phi_\sigma:\mathbb D^n \to X$ ...
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 ...
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 (...
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 ...
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 ...
4
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1answer
1k views

Covering of a CW-complex is a CW-complex

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration given ...
2
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1answer
105 views

Exercise Rotman 8.33 Algebraic Topology CW complex structure

If $(X,E)$ is a CW complex then so is $X \times I$. I have constructed the characteristic maps: We could give a cell decomposition of $X \times I$ by $E'' := \{ e \times a^0, e \times b^0 , e ...
4
votes
3answers
517 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
4
votes
1answer
111 views

$Y=\prod_{n\in\mathbb{N}}X$.

Let X be a compact CW-complex. The infinite cartesian product $Y=\prod_{n\in\mathbb{N}}X$ is a compact topological space, and as CW-complex It should have a finite number of cells. But in the CW-...
4
votes
1answer
110 views

Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)

I have a question about a proof in John Lee's Introduction to Topological Manifolds (5.22). Given CW-complex $X$ with skeletons $X_n$ and open cover $\left(U_\alpha\right)_{\alpha\in A}$, we ...
2
votes
1answer
659 views

Compact subset of CW complex

Let $X$ be a CW complex. Prove that a set $A\subseteq X$ is compact if and only if $A$ is closed and has a nonempty intersection with finitely many open cells of $X$ only. What I know: compact means ...
1
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1answer
264 views

Show that any cell complex is homotopy equivalent to a CW complex

We will give the definitions of cell complex and CW complex we use at the end of the post. Briefly speaking, "in a cell complex you don't have to glue cells in the order of their dimension, whereas ...
51
votes
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. ...
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
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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 ...
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 ...
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?
5
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1answer
301 views

Understanding construction of open nbds in CW complexes

I don't understand the construction of open nbds $N_\epsilon(A)$ of $A$ in a CW-complex $X$ given in page 522 of Hatcher's Book. Since the book is available for free online, I'll just copy the entire ...
5
votes
3answers
1k views

What is the difference between CW-complex and Cellular complex? [closed]

Is every CW-complex is a Cellular space? Is its converse true? If it is true then what is the difference between them? We include the definition of CW-complex in algebraic topology given by Whitehed ...
4
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2answers
609 views

Product of CW complexes question

I am having trouble understanding the product of CW complexes. I know how to actually do the computations and all, I just don't understand how exactly it works. So here's my questions specifically: ...
7
votes
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 ...
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 ...
4
votes
2answers
2k views

CW complex is contractible if union of contractible subcomplexes with contractible intersection

Exercise 0.23 from Algebraic Topology by Hatcher reads: Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible. I have ...
4
votes
1answer
387 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood $N_{\...
3
votes
1answer
898 views

CW construction of Lens spaces Hatcher

I am working through Hatcher's book and I am having trouble while understanding the CW-complex structure of Lens spaces. It is on page 145. He proves it constructing it in an inductive process. I ...
6
votes
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 ...
5
votes
2answers
261 views

Formula relating Euler characteristics $\chi(A)$, $\chi(X)$, $\chi(Y)$, $\chi(Y \cup_f X)$ when $X$ and $Y$ are finite.

This is a followup to my question here. Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the ...
5
votes
1answer
673 views

Euler characteristic for CW complexes

Can someone help me of how to prove these two basic properties of Euler characteristics, but regarding finite $CW$ complexes. $a)$ If $A$ and $B$ are two subcomplexes of a finite $CW$ complex $X$, ...
4
votes
0answers
165 views

Is $Y \cup_f X$ a CW complex?

Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the inclusion $A \to X$. My question is, is $...
9
votes
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 ...
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 ...
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, ...
4
votes
1answer
527 views

Try to generalize a problem in Hatcher: finite vs. infinite CW-complexes

While solving a problem in Hatcher I got this doubt in my mind, In the 2nd chapter (Homology) Hatcher asked us to prove the following question... If $X$ is a finite dimensional CW-complex then, a) If ...
3
votes
1answer
774 views

Quaternion Projective Space $\mathbb HP^n$ and Octonionic Projective Space $\mathbb OP^n$

$\mathbb{R}P^n$ and $\mathbb{C}P^n$ can be built as CW complexes. $\mathbb RP^n = e^0 \cup e^1 \cup \cdots \cup e^n$, $\mathbb CP^n = e^0 \cup e^2 \cup \cdots \cup e^{2n}$. Is there an analogous ...
3
votes
1answer
111 views

Bruhat-Tits Building of $PGL_3$: What does it look like?

I am very new to this topic, and my background is mostly in graph theory and basic algebra. What I want for now is to understand the structure of the dimension $2$ complex $\mathcal{B}(PGL_3(K))$ ...
3
votes
1answer
125 views

A non-positively curved cube complex that admits a local isometric embedding into a Salvetti complex is special.

I am trying to prove the following: "A non-positively curved cube complex $X$ that admits a local isometric embedding (that maps cubes to cubes) into the Salvetti complex of some right-angled Artin ...
3
votes
1answer
1k views

The motivation of weak topology in the definition of CW complex

Background A CW complex is a Hausdorff space and it is the union of its some of its subsets called cells, and cells are homeomorphic images in $X$ of some closed $k$-balls. The weak topology of a CW ...
2
votes
2answers
3k views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
2
votes
3answers
463 views

How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?

If A is the path connected CW complex and X is the new CW complex made by attaching a single 1-cell, is it true that the fundamental group of X is the same as that of A? I've tried to justify this ...
2
votes
2answers
261 views

If X is a simplicial complex, are the complex of simplicial chains and the one of cellular chains of X identical?

Let $X$ be a simplicial complex. The simplicial chain complex $C_*(X)$ is given by: $$ C_k(X) := \{\mathbb{Z}-\textrm{combinations of oriented simplices in X, with the identification }\sigma = - \...
0
votes
1answer
289 views

Is an image of a cellular map CW-complex?

Let $X$ and $Y$ be two CW-complexes and $f:X\longrightarrow Y$ be a cellular map. Is $f(X)$ a CW-complex? If the answer is no, then under what conditions $f(X)$ is a CW-complex? Thanks in ...
0
votes
1answer
485 views

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\...