Tagged Questions

2
votes
0answers
167 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
5
votes
1answer
79 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 ...
1
vote
1answer
65 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 ...
3
votes
2answers
110 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
3
votes
0answers
90 views

fundamental theorem of algebra proof

I have seen some proofs of the fundamental theorem of algebra using algebraic topology. But I have seen an exercise to prove the theorem by defining this map $f_t: S^1\longrightarrow S^1$ by $f_t(z) ...
1
vote
1answer
38 views

Every absolute retract (AR) is contractible

This is homework. I need to show that every AR is contractible. All I can basically do here is list definitions: A space $Y$ is AR if: $X$ is metrizable, $A$ is closed subset of $X$ and $f: A ...
1
vote
1answer
42 views

Homotopy equivalence of two spaces, homework

I need to prove that spaces $\mathbb R^2 \setminus \{e_1,-e_1\}$ and $S(e_1,1)\cup S(-e_1,1)$ are homotopy equivalent. $e_1$ is basis vector $(1,0)$ and $S(e_1,1)$ is a sphere centered on $e_1$ with ...
0
votes
0answers
42 views

The inclusion into the mapping cocylinder need not be a cofibration

The mapping cocylinder of a map $f:X\rightarrow Y$ is given by $N_f=\{(x,\beta)|f(x)=\beta(0)\}\subseteq X\times Y^I$. $f$ factors through a homotopy equivalence $j:X\rightarrow N_f$ given by ...
1
vote
0answers
82 views

homotopy groups of mapping space

I got this homework problem: $X,Y$ finite CW-complexes with $\dim X=m$ and $Y$ is $n$-connected. Prove that $\pi_k(map(X,Y))=0$ for all $k \le n-m$. Thanks for the help!
4
votes
0answers
54 views

Contractible and Compact space can be contained in an open set after time $t_0$?

$X$ is a topological space that is contractible and compact. Show that if $U$ is an open set in $X$ containing $x_0$ then there exists $t_0<1$ so that $H(x,t)∈U$, for all $x∈X$, and all $t_0≤t≤1$. ...
0
votes
1answer
41 views

homotopy of circles

Consider the circle with center in $(0,0)$ and radius 1 and the circle in $(2,0)$ and radius 1.5 in the plane. Are they homotopic (a) if we remove the origin, (b) if no point is removed? I think that ...
6
votes
1answer
502 views

Homotopy equivalence of universal cover

As part of am exam question (Q21F here), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, ...