Questions tagged [cw-complexes]

For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

150 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
15
votes
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 ...
8
votes
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
votes
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
votes
0answers
195 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
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 ...
7
votes
0answers
239 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
votes
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
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
0answers
475 views

Using the constructive definition of a CW-complex to prove that it is Hausdorff.

In class, I was given a constructive definition of a CW-complex and told that it's an easy exercise to prove that it is Hausdorff. I included the definition given in class below; We set $X^0 = \...
5
votes
0answers
92 views

Is $SU(n)$ a CW group?

A CW-group $G$ is a topological group with a CW structure such that both the multiplication and the inversion are cellular. It is known that $O(n)$, $U(n)$, and $Sp(n)$ can be given CW-group ...
5
votes
0answers
178 views

Is $\mathbb{S}^1 \wedge E$ a cofinal subspectra in $\Sigma E$?

I'm following the proof of Switzer's "Algebraic Topology and Homotopy" of the known result Theorem. Let $E$ be a (CW-pre)spectra. There is a natural (up to homotopy) homotopy equivalence $E \wedge \...
5
votes
0answers
316 views

The set of homotopy classes of maps is in bijection with homomorphisms between homotopy groups

My question comes from the fact that $\langle K(G,n),K(H,n)\rangle = \text{Hom}(G,H)$. The only proof of this fact that I know uses $\langle X,K(H,n)\rangle = H^n(X,H)$. However, this way of ...
4
votes
0answers
124 views

Cup product of $H^{\bullet}(\mathbb{R}P^{2};\,\mathbb{Z}/2)$ in terms of cellular cohomology

Considering the CW complex structure on $\mathbb{R}P^{2}$ consisting of one $0$-cell, one $1$-cell and one $2$-cell. Then the cellular chain complex of $\mathbb{R}P^{2}$ in $\mathbb{Z}/2$-coefficient ...
4
votes
0answers
196 views

Computing fundamental group. Using Van Kampen? Visualization of a space

I'm trying to compute the fundamental group (using Van Kampen) of a space which appears when identifying the disjoint boundaries of a 3-manifold with boundaries. My knowledge of 3-manifolds is none ...
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 $...
4
votes
0answers
230 views

On the path-connectedness of $X_{i} \setminus X_{i-2}$

Suppose $X$ is an $n$-dimensional regular CW-space (a space with a regular CW decomposition). What are the weakest sufficient conditions required on $X$ to ensure that all regular CW decompositions $(...
4
votes
0answers
109 views

Converting a space to a simplicial set

Given a compact manifold and a Morse function, we obtain a convenient cellular decomposition. But suppose I have a (finite dimensional, possibly singular) space $X$ that is not a manifold and I wish ...
4
votes
0answers
458 views

Homotopy of a CW complex

I have a CW complex constructed as follows: (The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein ...
3
votes
0answers
58 views

Can a CW complex have a subset of cells that is itself a CW complex, yet is not a subcomplex of the larger complex?

Let $(X, \mathcal{E})$ be a CW complex and $\mathcal{E'} \subseteq \mathcal{E}$ be a subset of cells such that $\mathcal{E'}$ is a CW decomposition of the space $X' = \bigcup_{e \in \mathcal{E'}} e \...
3
votes
0answers
30 views

Subspace topology in CW complexes

A subset $U$ of a CW complex is open iff for any closed cell $C$, the set $U \cap C$ is relatively open in $C$. My question is, can one characterize the subspace topology a similar way? That is, (when)...
3
votes
0answers
51 views

Relative Mapping Cylinders

Let $A$ be a nonempty topological space, and suppose $X,Z$ are topological spaces that contain $A$ as a subspace. Thus we have two pairs $(X,A)$ and $(Z,A)$. Then let $f:Z\to X$ be a continuous map ...
3
votes
0answers
34 views

How to visualize and explain complicated 3-dimensional structures?

I'm studying this example of nonshellable but constructible 3-ball on 10 vertices and 21 facets. Other than just staring at the pictures and hoping that they would eventually make sense, is there any ...
3
votes
1answer
48 views

Free resolution of a group $G$ and the chain complex of the universal cover of $K(G,1)$

Consider a group $G$ having a finite, free resolution $C_*(G)\to \mathbb Z$ over the group ring of $G$. I want to understand why this resolution may be viewed as the equivariant chain complex of the ...
3
votes
1answer
223 views

Homotopic maps and attaching spaces

Let $f, g : \mathbb{S}^{n-1} \to X$ be two continuous maps from the sphere into a compact and Hausdorff space $X$. I want to show that if $f$ and $g$ are homotopic, then attaching an $n$-cell to $X$ ...
3
votes
0answers
48 views

CW approximation of a topological group

Suppose we have a topologiacal group $G$. How do we construct a CW-complex $\bar{G}$ which is also a topological group, such that $\bar{G}\to G$ is a weak equivalence?
3
votes
0answers
67 views

Graph map terminology

By a directed multigraph, I mean a tuple $G= (V,E,i,t)$, where $V$ is a set of "vertices," $E$ is a set of "edges," and $i:E\to V$ and $t:E\to V$ are functions giving the "initial" and "terminal" ...
3
votes
0answers
146 views

Request for Details of Proposition 0.16 in Hatcher's Algebraic Topology

In Proposition 0.16 of Hatcher's Algebraic Topology, Hatcher claims that $ X^n \times I $ is obtained from $ (X^n \times \{0\}) \cup ((X^{n-1} \cup A^n) \times I) $ via attaching copies of $D^n \times ...
3
votes
0answers
179 views

Brief summary of simplicial, CW and manifold notions

I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a ...
3
votes
0answers
60 views

Not 1-dimensional homological equivalent of the circle

The questions origins from this problem and my incorrect answer to it. I'm trying to correct it, but it turned out that the topological space - that I need to do it straightforward - has very specific ...
2
votes
1answer
52 views

Pulback of a map to a pushout / pullback of cells in a fibration

Given a fibration $f:X \to B$ of CW complexes, it makes sense to guess that the pullbacks of a cell of $B$ will be a cell for $X$. That is, let $B_p$ be the $p$-th skeleton of $B$ and $X_p$ the ...
2
votes
0answers
50 views

Why the category of CW-complexes is not stable?

I am studying homotopy theory and I would like to understand better what it means for a category to be stable. For instance, the book I'm studying says that "the categories CW∗ and CCh+ have very ...
2
votes
0answers
40 views

Explicit cell structre of Torus with 3 points identified as 1

I have this problem "Let $X$ be the space obtained from the torus $S^1\times S^1$ by identifying three distinct points to one point. Find an explicit cell structure on $X$." It is a well-...
2
votes
0answers
50 views

A CW complex may have any point as a $0$-cell?

I have two questions about CW complexes. Let $X$ be any CW complex, and let $x$ be any point in $X$. Then may we assume that $x$ is always a $0$-cell of $X$? More precisely, is there a CW pair $(Y,y)...
2
votes
1answer
114 views

What are the cells of a simplicial complex?

I am trying to understand a bit the concept of CW complexes. It seems that simplicial commplexes are cell complexes as well. I would have said intuitively that in that case, the cells $e_{\alpha}$ are ...
2
votes
0answers
64 views

Definition of mapping telescope

In Kochman's stable homotopy theory, pg 121 prop 4.24 We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge ...
2
votes
1answer
133 views

CW complexes are the cofibrant objects in the Quillen model structure on Top?

If $J$ is a class of maps in a category, the $J$-cellular maps are by definition transfinite compositions of pushouts of coproducts of maps in $J$. Now if $J$ denotes the family of inclusions $S^{n-...
2
votes
0answers
140 views

Hatcher Exercise 3.2.16

Hatcehr Exercise 3.2.16. Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$ contain no elements of order a power of a given prime $p$, then the same is ...
2
votes
0answers
142 views

Map induced in homology by inclusion

Using Mayer-Vietoris sequence often one needs to know which map is induced in homology by an inclusion, let me try to explain with an example: Let's take the real projective plane with a hole i.e. $X=...
2
votes
0answers
158 views

Eilenberg-Zilber Theorem Proof

Let $X, Y$ be CW complexes and the following proof (from J. P. May's "A Concise Course in Algebraic Topology"; see page 102. Here the full pdf document: https://www.maths.ed.ac.uk/~v1ranick/papers/...
2
votes
0answers
43 views

Restriction of Attatching Map for CW Complexes Closed

A CW complex $X$ is a Hausdorff space and can be interpreted as a colimit $X = colim_k X_k$ of cells $X_k $ which satisfy can inclusion relations $X_{k-1} \subset X_k$ for each $k$ and are defined ...
2
votes
1answer
108 views

The space obtained by attaching a subspace of a topological space to itself.

Just asking a special case of attaching, it seems pretty vague to me. Let $Y$ be a space and $X$ be its subspace. Then the space obtained by $Y$ attaching its subspace $X$ via $\iota$ (the embedding) ...
2
votes
0answers
70 views

Weak product of Eilemberg MacLane spaces

I'm studying some homotopy theory of topological monoids from the book Algebraic Topology from a Homotopical Viewpoint. I'm trying to understand the corollary below. I'm stuck on the first claim of ...
2
votes
0answers
70 views

Using Homotopy Extension Propperty to prove that $S^{\infty}$ is contractible

I know that the question: ''Is $S^{\infty}$ contractible?'' has already been asked, but none of those post seem to answer it the way I need (Maybe they do, and my lack of knowledge prevents me from ...
2
votes
0answers
98 views

CW complexes are compactly generated

I am looking for verification and/or criticism of a proof I have constructed. The problem is to show that every CW complex is compactly generated. A topological space $X$ is $compactly$ $generated$ if ...
2
votes
0answers
351 views

Show that a CW-complex is contractible without Whitehead theorem

I was trying to solve the following problem: Let $X$ a CW- complex such that the inclusion $X_{n-1} \hookrightarrow X_n $ are null-homotopic, then $X$ is contractible. I already know that Whitehead ...
2
votes
0answers
58 views

Approximation by finite complex

Let $X$ be a simply connected space such that $H^k(X)$ is finitely generated for every $k$ and $H^k(X)=0$ for $k>n$. How can I find a finite $n$-(CW) complex $K$ homotopy equivalent to $X$? The ...
2
votes
0answers
92 views

Working out a proof of how to calculate $H_k(\mathbb{R}\mathrm{P}^n)$

The theoretical setting Assume $ X_{-1}:=\emptyset, X_0 \subset X_1 \subset \dots \subset X\;$ is a CW-complex with pushouts $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-...
2
votes
0answers
68 views

$G$ acts freely and cocompactly on $Y$ by permuting the cells, then $G$ is a factor group of $\pi_1(Y/G)$?

Let $Y$ be a connected CW-complex and $G$ a group acting freely on $Y$ by permuting the cells. We assume the action on $Y$ to be cocompact so that $X = Y/G$ is a finite CW-complex. Then how to see $G$ ...
2
votes
0answers
136 views

Homology of a the Orbit Space of a “Nice” Action of a Group on a CW-Complex

Let $X$ be a $CW$-complex and let $G$ be a group acting "cellularly" on $X$, that is, for each $g\in G$, the induced homeomorphism $\phi_g:X\to X$ takes cells to cells. Also assume that the action is ...
2
votes
0answers
122 views

isomorphic relative homolog groups of CW-complexes (using excision)

Consider pairs $(X,A),(Y,B)$ of $CW$-complexes, i.e. $X$ is a CW-complex and $A\subseteq X$ a CW-subcomplex and the same for $(Y,B)$. I want to know the following: How to prove in singular (...