Questions tagged [cw-complexes]

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

Filter by
Sorted by
Tagged with
1 vote
1 answer
49 views

"Introduction to Topological Manifolds" John Lee, Theorem 10.15

How do I know such a point $v$ below is guaranteed to exist? Wouldn't 3 closed disks side by side of radius 1, centered at (0,0), (2,0), and (4,0) in $\mathbb{R}^2$ be a connected finite CW complex? ...
SBJ's user avatar
  • 161
4 votes
1 answer
41 views

Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
Fabio Neugebauer's user avatar
-1 votes
1 answer
44 views

Is $X\times \{0,1\}$ in the unreduced suspension contractible?

I am trying to prove that the unreduced suspension of a CW complex $X$ is a CW complex and I am using the idea of the relative CW complexes given here Suspension of a CW complex my idea (in terms of ...
Emptymind's user avatar
  • 1,977
0 votes
0 answers
59 views

Fundamental group of the complementary of a cell complex

I am trying to understand the proof page 23 of this document which provides a method for computing fundamental group of the complement of a cell complex. Im doing an internship and trying to ...
Feffouu's user avatar
2 votes
0 answers
74 views

Partitioning Cellular Prism Operator?

In singular homology, we have the prism operator $P : C_{n}(X) \rightarrow C_{n+1}(Y)$ between singular chains, and for a (singular) $n$-simplex $\sigma$, $P(\sigma)$ decomposes into $(n+1)$ simplices ...
wanderer's user avatar
  • 255
4 votes
1 answer
40 views

Prove that removing an open 2-cell from $S^2$ results in a contractible space

Let $X$ be a cellular decomposition of $S^2$. I want to show that if $r\in X^{(2)}$ then $X\setminus \text{Int}(r)$ is a contractible space. I don't know much topology so I don't know if this it ...
Hernán Ibarra Mejia's user avatar
1 vote
1 answer
60 views

Algorithm for a simplicial presentation complex

Let $G$ be a finitely presented group with a finite presentation. Given this presentation, I want to construct via an algorithm a finite, $2$-dimensional simplicial complex $X$ with fundamental group $...
The_Rookie's user avatar
0 votes
1 answer
40 views

CW complex - closure finiteness and weak topology

Well, this is embarassing. To be honest, during my PhD, I haven't really bothered too much regarding the topology of CW complexes. Back then, I understood the first few pages of Hatcher's book (the ...
wanderer's user avatar
  • 255
1 vote
1 answer
66 views

An example of something that is not a subcomplex. [closed]

I was reading this part in Allen Hatcher book : But I did not understand it, can someone explain it to me with drawing please? Edit: I need a picture for the last 3 lines.
Intuition's user avatar
  • 3,269
1 vote
1 answer
80 views

Is the adjunction space of two Hausdorff spaces also Hausdorff?

I was reading the definition of CW-complex in terms of pushouts given by Lück's Algebraische Topologie: Homologie und Mannigfaltigkeiten (Chapter 3). It is stated (though not proven) that such a ...
Julius Maximus's user avatar
0 votes
0 answers
52 views

Question regarding Hatcher Exercise 0.24

The excercise wants us to show that for CW complexes $X$ and $Y$ with $0$-cells $x_0$ and $y_0$, we have a homeomorphism between the two spaces $$X*Y/(X*\{y_0\} \cup \{x_0\}*Y) \cong S(X \wedge Y)/S(\{...
KSAKY's user avatar
  • 121
2 votes
1 answer
58 views

Hatcher Theorem 1B.8 / Proposition 1B.9 concluding that a map inducing the identity is homotopic to the identity?

In introducing the concept of $K(G, 1)$, Hatcher's Algebraic Topology proves the theorem that the homotopy type of a CW complex $K(G, 1)$ is uniquely determined by $G$, by citing a proposition and ...
I Eat Groups's user avatar
0 votes
1 answer
75 views

CW Complex Decomposition of $S^{n}$

I'm currently delving into the foundational concepts of algebraic topology and came across the definition of CW complexes in Hatcher's "Algebraic Topology" (Chapter 0, p. 5): (1) Start with ...
whymgang's user avatar
3 votes
1 answer
127 views

Want a simple construction of a relative CW complex (something like a torus) analogous to regular CW Complex constructions.

I think I'm getting burnt out, as I'm having trouble understanding Spanier's definition of a relative CW complex (from his Algebraic Topology): A relative CW complex $\big(X, A \big)$ consists of a ...
Nate's user avatar
  • 780
1 vote
1 answer
50 views

how to understand a simplicial complex as a CW structure

I learned in my algebraic topolgy course that $X$ is a CW complex if there is a filtration $$ \emptyset = X_{-1} \subset X_0\subset \dots\subset X_n \subset \dots \subset X $$ such that $X$ is the ...
user1072285's user avatar
0 votes
1 answer
52 views

S : ∞-Grpd₀ ⭢ ∞-Grpd₀ such that Homology of X is the homotopy of SX?

Let $X$ be a based CW-complex, and write ∞-Grpd₋₁ for the category of based CW-complexes. I am wondering if there is some operation S or functor on spaces, remaining in the homotopy category, which ...
user avatar
1 vote
2 answers
96 views

"Open" Cell? (Hatcher and Husemoller)

Have $\mathbb D^n$ be some n-dimensional unit disk. Have $\partial S$ denote the boundary of some space $S$. Let infix $(-)$ denote a difference between collections of objects. In Dale Husemoller's ...
Nate's user avatar
  • 780
0 votes
1 answer
97 views

A simple question about Hatcher's notation (abuse?) in Algebraic Topology when defining coproducts.

This is a straightforward question, but not one that Hatcher clarifies: Hatcher likes to write the disjoint union of topological spaces with infix $\coprod $. This works well enough, but there's a ...
Nate's user avatar
  • 780
2 votes
1 answer
85 views

Are transfinite cell complexes the same as the usual cell complexes?

Let $\alpha$ be an ordinal, considered as a set of the ordinals less than it. Aditionally suppose we have a function $h:\alpha+1 \to \mathbb N$. Suppose for each $i \leq \alpha$ we have a topological ...
Fernando Chu's user avatar
  • 2,541
8 votes
2 answers
181 views

Lifting a map from the quotient by a contractible topological group

Let $X$ be a topological space, $G$ be a contractible topological group acting freely and continuously on $X$, and $Y$ be the quotient. Let $A$ be a finite-dimensional CW-complex, and $f: A \...
Taras's user avatar
  • 321
2 votes
0 answers
60 views

Question about an alternative description of the topology on CW complex from Hatcher

This is a statement from Hatcher's Algebraic Topology that I can't completely understand. So we define a CW complex $X$ inductively starting with a discrete set $X^0$ the $0$-cells of $X$. Then ...
nomadicmathematician's user avatar
0 votes
1 answer
74 views

show that a quotient space is a CW complex.

Here is what I am thinking about: I want to show that the quotient space obtained from a polygon $P$ by identifying some of its edges together in pairs is a CW complex. I can assume without proof that ...
Emptymind's user avatar
  • 1,977
0 votes
1 answer
74 views

The attaching maps and the characteristic maps of a new CW structure.

I know that the sphere $S^n$ has the structure of a cell complex with just two cells, $e^0$ and $e^n,$ where the $n$- cell being attached by the constant map $S^{n-1} \to e^0.$ I also know that ...
Hope's user avatar
  • 95
4 votes
0 answers
54 views

Automorphisms of CW complexes and fixed points

Let $X$ be a CW complex, and let $F:X\to X$ be an homeomorphism that sends each cell onto some cell; notice that we could say that $F$ is an automorphism of the CW complex since it preserves its cell ...
mathplayer's user avatar
0 votes
1 answer
70 views

Plus construction and classifying space

Suppose $G$ is a perfect group, and let us consider $BG$ and its plus construction (We consider $BG$, the classifying space, by the nerve construction). By following Hatcher’s book (Proposition 4.40, ...
wanderer's user avatar
  • 255
0 votes
1 answer
47 views

Boundary operator $\partial_n$ equal to composition $i_* \circ \delta$

I know that for a CW-complex $X$ with skeletons $X^n, n \geq 0$, one can identify the cellular chain group $C_n^{\text{cell}}(X)$ with the singular homology group $H_n(X^n, X^{n - 1}).$ What I don't ...
Minerva's user avatar
  • 153
2 votes
0 answers
31 views

CW complex structure on manifolds with certain dimensional cells

I started reading Morse theory and I learnt one of the fundamental theorem which says that any smooth closed manifold admits a CW complex structure. I had a following question: Given a smooth, closed ...
Alexander93's user avatar
1 vote
1 answer
76 views

Plus construction is functorial

Given a nice map (cellular map) $f : X\rightarrow Y$ between CW complexes $X$ and $Y$, how is $f^{+}$ defined between their plus constructions? Going through my old notes, I've learnt that the plus ...
wanderer's user avatar
  • 255
0 votes
0 answers
54 views

Inclusion of skeleton into CW complex gives isomorphism in relative homology

This question is actually formulated for cohomology but I think the proof will be dual. Let $H_\bullet$ be a general homology theory satisfying dimension axiom and union axiom. Let $(X,A)$ be a ...
Gargantuar's user avatar
0 votes
2 answers
54 views

Proof that a compact subset of a CW complex is contained in a finite subcomplex

The following is from Theorem 5.14 of John Lee's Introduction to Topological Manifolds. Let $X$ be a CW complex. A subset of $X$ is compact if and only if it is closed in $X$ and contained in a finite ...
nomadicmathematician's user avatar
0 votes
0 answers
40 views

Algorithm to recognize spherical abstract polytopes

A finite abstract polytope of rank 3 (an abstract polyhedron) consists of adjacency data for a collection of polygonal faces and their shared edges and vertices. This data is sufficient to uniquely ...
Karl's user avatar
  • 11.5k
0 votes
1 answer
55 views

Gluing 2-cell and Euler-Poincaré characterstic

Let $Y$ be a $n$-dimensional $CW$ complex. This means that $Y$ is a cellulation of topological spaces $$Y=Y_n\supseteq Y_{n-1}\supseteq\dots\supseteq Y_0$$ Where $Y_0$ is a non-empty set of points and ...
Gabriele's user avatar
0 votes
1 answer
37 views

A specific cell complex

We construct a cell complex by attaching the boundary of a two dimensional disk $D$ to $S^1$ by $z\to z^n$ ($n>2$). This cell complex seems to be closed, compact and connected. But it's ...
max101's user avatar
  • 1
3 votes
0 answers
130 views

Examples using this long exact sequence of cohomology [duplicate]

I began reading this paper (also provided below), and I would really appreciate if someone could help me with the premises behind the first sentence. I have not been able to find the given long exact ...
June in Juneau's user avatar
2 votes
0 answers
35 views

Why $EG \to EG/G$ is a fibration?

I am trying to prove that for any group $G$, if $EG$ is a contractible CW-complex on which $G$ acts freely, and for which the action of $G$ is cellular and free on the cells of $EG$, then the ...
Antoine Rodrigues's user avatar
1 vote
1 answer
67 views

CW structure on 2-sphere with $K_5$ and $K_{3,3}$ as 1-skeletons with face edges self-identified

The following is an exercise from Hatcher's Algebraic Topology: Suppose we build $\mathbb S^2$ from a finite collection of polygons by identifying edges in pairs. Show that in the resulting CW ...
D Ford's user avatar
  • 3,997
1 vote
1 answer
79 views

A detail in homology of CW-complexes

Here is a part of my notes in the algebraic topology class: Let $A$ be a space, $k\geqslant 2$. $f:S^{k-1}\rightarrow A$ be a map. Then consider the adjunction space $$ A\cup_f e_k=A\cup_f CS^{k-1}$$ ...
Zoudelong's user avatar
  • 175
0 votes
0 answers
49 views

Is the first term of this sequence trivial?

I have a CW-complex $X$ with a closed subset $C$ and its complement $U$. Like in this similar question, I am working with the following long exact sequence of cohomology with compact supports: $$ \...
June in Juneau's user avatar
0 votes
0 answers
31 views

Higher connectedness of product of CW complex

Take $X,Y$ two pointed CW-complexes, with $X$ $k$-connected, $Y$ $l$-connected, we may also assume that $X$ or $Y$ is locally compact. I'm trying to show $(X\times Y, X\vee Y)$ is $(k+l+1)$-connected. ...
t_kln's user avatar
  • 1,048
2 votes
1 answer
160 views

Computation of $H^∗ (Σ\mathbb RP^n , \mathbb F_2)$ as graded $\mathbb F_2$-algebras

I'm trying to compute $H^*(\Sigma\mathbb{R}P^n,\mathbb{F}_2)$ as a $\mathbb{F}_2$ graded algebra. I know that the cup product is 0 since since it is for all suspensions, but I'm not sure how to use ...
bml64's user avatar
  • 627
3 votes
1 answer
87 views

Does homotopy type of a map $S^1 \to RP^1$ defines a homotopy type of $(D^2, S^1)\to (RP^2, RP^1)$?

Consider a CW structure on $\mathbb{R}P^2$ with one 2-cell, one 1-cell $\mathbb{R}P^1$, and one 0-cell. Let $f \colon (D^2, \partial D^2) \to (\mathbb{R}P^2, \mathbb{R}P^1)$ be a map such that $f \big|...
Alex's user avatar
  • 141
1 vote
1 answer
50 views

Pushfoward of a CW complex structure by a covering map

Let $p:Y\to X$ be a covering map. If $X$ has a CW complex structure, then we can give a CW complex structure on $Y$ so that $p$ becomes a cellular map, by lifting the characteristic maps (cf. Euler ...
user302934's user avatar
  • 1,698
1 vote
2 answers
125 views

Category whose classifying space is $S^n$

Suppose one wants a finite category $C_n$ (finite sets of objects and morphisms), for which $\left\vert NC_n \right\vert\simeq S^n$. I think one such $C_n$ can be built as follows : Take $\partial \...
t_kln's user avatar
  • 1,048
2 votes
1 answer
76 views

A CW-decomposition of $S^n$

Here is one of my homework in the topology class: Show that $S^n=\{x_0\}\cup_{C_{x_0}} e^n$, where $C_{x_0}:S^{n-1}=\partial e^n\rightarrow \{x_0\}$ is the constant map. My attempt: We construct a map ...
Zoudelong's user avatar
  • 175
0 votes
0 answers
45 views

Prove $X$ has the weak topology where $p:X\to Y$ is a covering map.

This question has been answered here, but the method used is different than the one I learned in my algebraic topology course, which I believe I should use. I showed that given that $Y$ is a CW ...
user13121312's user avatar
1 vote
1 answer
100 views

Hatcher 4.2.19 homotopy group of skeleton of $K(G,1)$

Let $X$ be a $K(G,1)$ which is a CW complex. We want to show that $\pi_n(X^n)$ is free (abelian when $n\geq 2$ where $X^n$ is the $n$ skeleton of $X$. I thought I could choose any model for a $K(G,1)$ ...
DevVorb's user avatar
  • 1,347
0 votes
0 answers
80 views

Homology of $S^1\times \mathbb{RP}^2$ without using Kunneth's Theorem

I am trying to compute the homology of $S^1\times \mathbb{RP}^2$ for all $n\geq 0$. I read online that Kunneth's Theorem about homology of product spaces might be useful, but unfortunately, we haven't ...
user13121312's user avatar
2 votes
1 answer
94 views

Weak Homotopy Equivalence Induces Isomorphisms on Homology

Hatcher 4.21 says that a weak homotopy equivalence $f : X\to Y$ induces isomorphisms $f_* : H_n(X; G)\to H_n(Y; G)$. In the proof he says that by replacing $Y$ by the mapping cylinder $M_f$ and ...
J Crane's user avatar
  • 53
1 vote
0 answers
44 views

Homotopy Long Exact Sequence Coming From the Mapping Cylinder

Hatcher defines the mapping cylinder $M_f$ of a map $f : Z\to X$ as the quotient space of the disjoint union of $Z\times I$ and $X$ under the identifications $(z, 1) \sim f (z)$. My question is that ...
J Crane's user avatar
  • 53
0 votes
1 answer
38 views

Subcomplex of the standard $n$-simplex having nonzero $H_{n-1}$

This is an additional exercise of Hatcher's Algebraic Topology book (link: https://pi.math.cornell.edu/~hatcher/AT/AT-exercises.pdf, section 2.1). Let $X$ be the standard $n$-simplex with its natural $...
user302934's user avatar
  • 1,698

1
2 3 4 5
18