Questions tagged [cw-complexes]
For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).
772
questions
0
votes
1
answer
60
views
show that if a simply connected space has $H_2(X) = \mathbb{Z} \oplus \mathbb{Z}$ then it is homotopic to $S^1 \bigwedge S^2$
The problem is as follows
show that if a simply connected CW-complex has $H_2(X) = \mathbb{Z} \oplus \mathbb{Z}$ and $H_i(X) = 0$ for all $i \neq 2$ then it is homotopy equivalent to $S^1 \bigwedge S^...
- 503
1
vote
0
answers
28
views
Where can I get an alternative proof of this characterisation of weak equivalences?
Brayton Gray's book Homotopy Theory is the only book I've seen that states and proves the following theorem.
Lemma 16.17. Suppose $f: X \to Y$ is a base point preserving map. $f$ is a weak homotopy ...
- 1,572
0
votes
0
answers
31
views
the relation between well-pointed spaces and CW complexes
I was reading the second solution here:
Suspension of a product - tricky homotopy equivalence but the author said that the situation of well-pointed spaces is considered a more general situation than ...
- 2,007
1
vote
1
answer
56
views
Why does the $0$-skeleton have discrete topology in CW complexes?
I am reading about CW complexes. My book uses the following definitions.
Definition: A cell complex $X$ is a Hausdorff space which is the union of disjoint subspaces $e_{\alpha} (\alpha \in \mathcal{A}...
- 1,572
4
votes
2
answers
198
views
Understanding CW complex construction
Hatcher's construction of a CW complex is as follows (see page $5$ of Hatcher's Algebraic Topology):
(1) Start with a discrete set $X^0$ , whose points are regarded as $0$ cells.
(2) Inductively, form ...
- 10.1k
0
votes
1
answer
52
views
Questions regarding basic definitions on CW complexes?
I am reading about CW complexes. My book uses the following definitions.
Definition: A cell complex $X$ is a Hausdorff space which is the union of disjoint subspaces $e_{\alpha} (\alpha \in \mathcal{A}...
- 1,572
0
votes
0
answers
82
views
Homology group of sphere relative to the complement of finitely many points
Let $X=S^n, n>0$ and $A\subseteq X$ such that $X\setminus A$ contains finitely many points $x_1,...,x_m$. I need to show that the reduced homology $H_n(X,A)=\mathbb{Z}^m$.
I need to do this ...
- 533
0
votes
0
answers
82
views
Is a regular CW complex also a normal CW complex?
In the book "The Topology of CW Complexes" p.78,
A CW complex is regular if each closed cell is homeomorphic to a closed Euclidean n-cell.
A CW complex is normal if each closed cell is a ...
- 11
1
vote
0
answers
24
views
The definition of a maximal tree as a 1 dimensional cw-complex
In Hatcher's - a maximal sub-tree in a graph is defined as a contractible subgraph that reaches all the vertices.
Later he shows that this definition is equivalent to a cycle-free connected graph - ...
- 33
2
votes
1
answer
51
views
A way to construct a K(G,1) cw-complex for an arbitrary G
I am currently reading the appendix 1.B of Hatcher's. There he described a way to construct a $\Delta$-complex which is $K(G,1)$ for an arbitrary group $G$. Later in the chapter he used the fact that ...
- 33
2
votes
1
answer
52
views
Effects on CW-complexes by replacing attatching maps by homotopic ones
Let $X$ be a $n$-dimensional dimensional CW-complex and $\pi_k(X)=0$ for $ n \ge k >1$. Therefore obviously by construction every attaching map $f_i^k: S^k \to X^{(k)}$ of a $(k+1)$-cell $e_i^{k+1}$...
0
votes
1
answer
98
views
Kill a Homology class $[\beta] \in H_k(X)$ only by attaching $(k+1)$-cells
Let $X$ be a $n$-dimensional dimensional CW-complex build of skeleta $(X_0,X_1,...,X_k ,..., X_n) $. Let $[\beta] \in H_k(X)$ a non zero homology class which by definition can be represented by a ...
0
votes
1
answer
42
views
Suppose $X$ is obtained from $A$ by attaching $(n+1)$-cells. Then $(X,A)$ is $n$-connected.
I'm studying the book "Algebraic Topology" by Tammo Tom Dieck: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf
I don't understand the proof of proposition 8.5.1:
$\textbf{(8.5.1) ...
- 172
3
votes
2
answers
115
views
Relative cell complexes in the undercategory $B/\mathscr{M}$ are relative cell complexes in $\mathscr{M}$ - why must we also assume $B$ is a complex?
$\newcommand{\M}{\mathscr{M}}\newcommand{\I}{\mathcal{I}}\newcommand{\J}{\mathcal{J}}$The context: the following is claimed in J. May's "more concise algebraic topology".
We have some model ...
- 22.2k
0
votes
2
answers
67
views
Proving Invariance of CW Complex Dimension
John Lee's Introduction to Topological Manifolds defines the dimension of a CW complex $X$ as the largest dimension of a cell in $X$. The author claims "The fact that this well defined depends on ...
- 1,995
2
votes
0
answers
90
views
Modification of a CW-Complex $X$ with control over Homology groups
Let $X$ be a finite dimensional CW-complex, ie we endow it with CW structure such that there exist a $n_0>0$ such that there are no $n$-simplices for $n>n_0$. Let $[\beta] \in H_k(X, \mathbb{Z}),...
1
vote
1
answer
55
views
Uniqueness of Characteristic Maps in Cell Decomposition
Suppose $X$ is a topological space with a cell decomposition, that is, a partition $\mathcal{E}$ of $X$ into open cells of various dimensions such that whenever $e \in \mathcal{E}$ is an $n$-cell, ...
- 1,995
1
vote
1
answer
72
views
Connected one-dimensional CW complex and homotopy groups
Prove that connected one-dimensional CW complex X has $\pi_n(X)=0, n \ge 2$.
This is a problem from my exam, I tried to use the cellular approximation theorem but didn't solve it. Professor said it is ...
- 401
0
votes
1
answer
40
views
Transitivity of CW-pairs
If $(X,A)$ is a CW-pair with $A$ a subcomplex of $X$ and $(Y,X)$ is a CW-pair with $X$ a subcomplex of $Y$ is $(Y,A)$ a CW-pair with $A$ a subcomplex of $Y$?
4
votes
2
answers
49
views
Number of homotopy classes of maps $X\to Y$ of CW-complexes
Determine whether there are one, finitely many or infinitely many homotopy classes of maps $f\colon X\to Y$:
$X=S^1\times S^1,\quad Y=S^3$
$X=S^1\times S^1,\quad Y=S^1$
$X=S^1\vee S^3,\quad Y=S^3$
...
- 732
0
votes
0
answers
57
views
$A\hookrightarrow X$ induces an injection $H_n(A)\hookrightarrow H_n(X)$
Show that if $(X,A)$ is a CW pair of dimension $n$ (so all cells of $X-A$ have dimension at most $n$) then the map $H_n(A)\to H_n(X)$ induced by the inclusion $A\to X$ is injection with image a direct ...
- 23
3
votes
1
answer
113
views
What is an example of a closed manifold whose universal cover is not of finite homotopy type?
Does anyone know of an example of a closed manifold whose universal cover does not have the homotopy type of a finite CW complex?
(If the universal cover is also compact, then it is of finite homotopy ...
- 405
1
vote
0
answers
40
views
Why are groups of type $F_2$ finitely presented?
There are some equivalent definitions for "finiteness properties", but let's define that $G$ is a group of type $F_n$ if it is the fundamental group of a CW complex $X$ whose $n$-skeleton is ...
- 81
1
vote
1
answer
64
views
Topology on a graph as a CW-complex
I want to make sure I'm correctly understanding the topology that a graph is given when thought of as a $1$-complex. From what I can tell, the topology should be generated by sets of the following ...
- 1,218
2
votes
1
answer
40
views
Suspension and reduced suspension
It is clear to me that
for a space $X$ , the suspension $SX$ is the quotient of
$X × I$ obtained by collapsing $X×\{0\}$ to one point and $X × \{1\}$ to another point. The motivating example is $X = S^...
- 3,911
0
votes
1
answer
97
views
CW complex for Möbius strip and its homeomorfisams
I have to find CW complex for Mobius Strip. I can find a CW complex for a Möbius strip with more cells (one 0-cells, two 1-cells and a single 2-cell). My cells are points $A=(0,0) \approx (1,1)$, part ...
- 401
0
votes
1
answer
65
views
Can we glue CW complexes to themselves? [closed]
My understanding: CW complexes are constructed by inductively attaching $n+1$ cells to an $n$-skeleton where a 0-skeleton consists of a discrete set of 0-cells (points).
What I fail to understand: ...
- 525
0
votes
0
answers
52
views
proving that the inclusions $i:X \to CX$, $j:Y \to C_f$, $j':Y \to M_f$ have the homotopy extension property
Let $X$ be a topological space, and $f:X \to Y$ a continuous map.
I want to show that the inclusions $i: X \to CX, \: j:Y \to C_f, \: j':Y \to M_f$ all have the homotopy extension property. Where $CX$...
- 717
0
votes
0
answers
33
views
For a cw-complex the pair $\left(X,X^n \right)$ is n-connected
Hello I am currently self-learning about (algebraic-)topology using Hatchers book.
My definition of $n$-connected is that $\pi_k\left(X,X^n\right)=0$ for all $1\le k \le n$.I do know about the ...
- 2,231
1
vote
1
answer
96
views
Prove that $\langle a,b\mid ab=ba^n\rangle$ is not isomorphic to $\langle a,b\mid ab=ba^{n'}\rangle$ when $n\neq n'$
For a positive integer $n$, let $X_n$ be be the quotient space $\left([0,1] \times S^1\right) / \sim$, where the equivalence relation $\sim$ is generated by
$$
(0, z) \sim\left(1, z^n\right), \quad \...
- 855
1
vote
2
answers
84
views
First differential in cellular homology
How do I compute the first differential of cellular homology in general for a CW complex $X$, without using for example that the first differential is zero if $X$ is path-connected and has one zero ...
- 11
1
vote
1
answer
35
views
Topological description for graph minors
In the following content, graphs are considered inside $ZFC$ as the $1$-dimensional CW-complexes deduced from its cardinal-valued (multigraphs may have cardinals larger than $1$ as entries) adjacent &...
- 1,814
0
votes
0
answers
40
views
Why is the Euler characteristic of a torus with four identified points equal to -2?
This does not any sense at all
If I take a torus, and take four points $x_1, x_2, y_1, y_2$ and identify $x_1$ with $y_1$ and $x_2$ with $y_2$, then I get a torus which two "singularities" ...
- 123
1
vote
1
answer
48
views
Definition of subcomplex of a CW-complex
I am studying Algebraic Topology from Hatcher, they define a subcomplex of a cell complex X as a closed subspace A of X such that A is a union of cells of X.
I understend this definition, but for me ...
- 172
1
vote
1
answer
69
views
Showing the shearing map between product CW space is a homotopy equivalence.
$\mathbf {The \ Problem \ is}:$ Let $(X,*)$ be a based,path-connected CW complex with a based, continuous $\mu: X\times
X \to X$ with $\mu(*,z)=z=\mu(z,*)$ for all $z\in X.$
Then , $X$ is a H-space ....
- 2,037
2
votes
0
answers
55
views
Using chain homotopy equivalences of two chain complexes, show two spaces are homotopy equivalent.
$\mathbf {The \ Problem \ is}:$ Let $X$ be a simply connected CW-space with base point as a single $0$-cell . Let $C$ be the reduced cellular chain complex of $X$ (quotient of cellular chain complex ...
- 2,037
3
votes
1
answer
81
views
How can we rigorously cellulate spaces we can't visualise? Difficulty finding nontrivial cellulations of the $n$-cube, $n$-torus
In the interests of understanding cellular homology better I decided to try to verify for myself the identity: $$\large H_k((S^1)^n)\cong\Bbb Z^{\binom{n}{k}}$$
Using cellular homology. To do that, I ...
- 22.2k
0
votes
1
answer
30
views
higher-degree attaching maps in cell complexes
I'm new to algebraic topology -- sorry if this is a silly question or covered in some obvious place.
I didn't find it in Hatcher or other questions on this site.
What I get so far:
Consider an $n$-...
- 381
3
votes
2
answers
87
views
In cellular homology, how should we define the degree of a map between $0$-spheres?
Cellular homology is one of many homology theories available for 'nice' spaces. There are doubtless many equivalent definitions for it, in increasing levels of abstraction, but I want to ask about the ...
- 22.2k
1
vote
1
answer
78
views
CW-complex of Möbius band without 0-cells
So proofwiki, and some other sites, claim the Euler-characteristic of the Möbius strip is 0-1+1=0.
Relying on the fact that a Möbius strip has no vertices, i.e. 0-cells.
However I can nowhere find a ...
- 53
4
votes
1
answer
83
views
Homotopy type of a simply connected CW space with all homology groups trivial after a certain stage.
$\mathbf {The \ Problem \ is}:$ Let, $X$ be a simply connected based CW space with $H_k(X)=0$ for all $k>c.$ Show $X$ is homotopy equivalent to a CW space $\Gamma X$ with no $t$-cells for $t>(c+...
- 2,037
1
vote
1
answer
72
views
Showing homotopy equivalence of identity map from graph topology to metric topology on the same set
To elaborate on the question:
Consider set X equipped with 1-dimensional CW complex topology ($\mathcal{T_{CW}}$) which we can think essentially as graph. Edges are thought of as intervals $[0,1]$.
...
0
votes
1
answer
43
views
CW structures compatible with fiber bundle structures
Suppose $F \to E \to M$ is a vector bundle whose base space $M$ and fiber space $F$ are both CW complexes. Let us call a CW structure on $E$ a “compatible with the bundle structure” if, for any cell $...
- 1,653
3
votes
1
answer
130
views
Cohomology class $ H^k(X; \pi_{k-1}(Y,y_0))$ as obstruction
I am following these notes http://scgp.stonybrook.edu/wp-content/uploads/2018/09/lecture-1.pdf
on obstruction theory.
Let $X$ be a CW complex and denote by $X^{(k)}$ its $k$-skeleton. Suppose
that $Y$ ...
0
votes
0
answers
15
views
Does gluing polyhedra together give you a CW complex?
If I take a finite collection of polyhedra and glue their faces together (potentially gluing two faces of the same polyhedron together) to get a closed 3-manifold, is this always a CW-complex? Can I ...
- 11
1
vote
0
answers
18
views
Is $\pi_2 (X)$ a projective $\mathbb{Z}\pi_1 (X)$-module when $\pi_1 (Z)$ is free?
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
- 2,585
0
votes
1
answer
62
views
Topology of the direct sum $\bigoplus_{n \in \mathbb N} \mathbb R$
Let $S$ and $P$ be the direct sum and direct product of countably many copies of $\mathbb R$. Clearly, $P$ should have the product topology. On $S$, we could consider two different topologies:
the ...
- 1,653
0
votes
1
answer
135
views
Hatcher 0.25: Wedge sum of suspensions of components of $X$ is suspension of $X$
I can't figure out how to do problem 0.25 from Hatcher's Algebraic Topology. It says
If $X$ is a CW complex with components $X_{\alpha}$ , show that the suspension $SX$ is
homotopy equivalent to $Y \...
1
vote
1
answer
52
views
Degree of maps of product CW complex
If we treat the points of $S^3$ as quaternions i.e. $S^3=\{y_0+y_1i+y_2j+y_3k \mid y_0^2+y_1^2+y_2^2+y_3^2=1\}$, let $X=\mathbb{R}P^3$ by identifying antipodal points of $S^3$. Then quaternion ...
- 337
0
votes
0
answers
71
views
Cell decompositions and cell complexes
Let me first state my definition of cell complexes.
Definition 1: Cellular decomposition of a topological space $X$ is a cover $\mathcal{E}$ of space $X$, equipped with a function deg: $\mathcal{E} \...
- 1,572