Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...
2
votes
0answers
19 views
graphs and homotopy extension property
If $T$ is a spanning tree of a graph $X$. How to prove that the pair $(X,T)$ has the homotopy extension property, without using the definition of CW complexes? I mean I don't need the general case ...
5
votes
1answer
58 views
The action of the group of deck transformation on the higher homotopy groups
This is for homework.
I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
0
votes
0answers
19 views
Is there a fibration whose base, total and fibre is a Moore space?
I want to find a example which base, total of fiber spaces is a Moore space.
I don't want trivial fibration.
For example the projection $p : M(G,n) \times M(H,n) \to M(G,n)$ and path fibration ...
6
votes
1answer
49 views
Can the rank of the homology group of an abstract simplicial complex be computed in polynomial time?
I want to write a function that does the following:
Input:
An integer $n$
A function $f$ that maps nonempty subsets of $\{1, \dots, n\}$ to "yes" or "no" such that (a) every singleton set gets ...
0
votes
0answers
67 views
Homotopy, retraction related question
How to solve the following problem:
$A$ is a strong deformation retract of $X$ such that exists a continuous function $α:X\to I$, $\alpha^{−1}({0})=A$.
(a) If $H:X\times I\to X$ is a homotopy (rel ...
6
votes
0answers
63 views
Visualize Fourth Homotopy Group of $S^2$
I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
4
votes
1answer
66 views
What's the point of spectra?
I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
2
votes
0answers
25 views
Naive question: Why $B:Mon\rightarrow Top^{*}, \Omega: Top^{*}\rightarrow Mon$ is an adjoint functor?
It is not clear to me why we have a bijection of the form $$Mor_{Top^{*}}(BY,X)\rightarrow Mor_{Mon}(Y,\Omega X)$$where $Mon$ is the category of topological monoids and $Top^{*}$ the based topological ...
1
vote
1answer
42 views
Covering infinite sheeted covering of torus
Suppose I have subgroup $H=\operatorname{span}\langle (a,b)\rangle\subset \pi_1(\mathbb{T}^2)=\mathbb{Z}^2$, where $a,b$ are integers where $(a,b)\neq(0,0)$. I know the covering space is $S^1\times ...
5
votes
1answer
95 views
What is the homotopy colimit of the Cech nerve as a bi-simplical set?
Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set
$$
F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
1
vote
1answer
52 views
$SU(n)$ is simply connected (proof without fibrations, $n>2$)
How to show that $SU(n)$ is simply connected for $n>2$ if I don't know about fibrations yet? For $SU(2) \cong S^3$ the fact is said to be known. For any matrix $A \in SU(n)$ there is a matrix $S ...
5
votes
2answers
32 views
elementary reference for Hopf Fibrations
I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded accessible book. ( I should probably say that I am just starting to learn ...
2
votes
1answer
27 views
Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?
I'm looking for a reference for the following result:
If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
1
vote
0answers
23 views
Mathematica code for solving MHD viscoelastic fluid flow [migrated]
Where might I find Mathematica code for solving MHD viscoelastic fluid flow using Homotopy Analysis method?
3
votes
0answers
37 views
How to show $\mathbb{S}^{2n-1}$ is a $\mathbb{S}^{n-1}$ bundle over $\mathbb{S}^{n}$ then $n=1,2$ or divisible by $4$?
I was asked to show
if $$\mathbb{S}^{n-1}\rightarrow \mathbb{S}^{2n-1}\rightarrow \mathbb{S}^{n}$$ holds. Then $n=1,2$ or divisible by $4$. The $n=1,2$ cases in the converse are verfied. I am ...
