3
votes
3answers
134 views

how should I show that it is wedge of infinite circles?

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A ...
1
vote
1answer
58 views

Prove fundamental group is the direct product

Suppose that $A$ is a retract of $X$ with retraction $r : X \rightarrow A$. Also suppose that $i_*(\pi(A,a))$ is a normal subgroup of $\pi(X,a)$. Prove that $\pi(X,a)$ is the direct product of the ...
2
votes
0answers
91 views

Fundamental group of CW-Complex only depends on 2-Skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1 (X)$ only depends on the $2$-Skeleton of $X$ for any $CW$-Complex $X$. I am very ...
0
votes
1answer
47 views

Need help on finding homotopy

Define a continuous map $\ell:(I,\partial I)\to (SO(3),1)$ by $\ell(t) = \left( \begin{array}{ccc} \cos 2\pi t & -\sin 2\pi t & 0 \\ \sin 2\pi t & \cos 2\pi t & 0 \\ 0 & 0 & 1 ...
1
vote
1answer
34 views

Non self-intersecting representatives in fundamental class

If $X$ is a Riemann surface with boundary $\partial X$ and $\pi_1(X,p)$ is its fundamental group, $p \in X$, then we shall call class $[\gamma] \in \pi_1(X,p)$ primitive (or generator) if it can not ...
1
vote
2answers
26 views

Seeking 'simple' space with specified homotopy

I am looking for a 'named' space $S$ such that $\pi_1(S) = \mathbb{Z}_2$ and $\pi_n(S) = \star$ (the one-point group) for all $n\geq 2$. Commentary: I know that the projective plane fits the first ...
1
vote
3answers
74 views

$\pi_1(X)\cong \mathbb Z_{p^n}$

If $p$ is a prime. Can one construct a space $X$ such that $\pi_1(X)\cong \mathbb Z_{p^n}$, for any $n\in \mathbb N$?
0
votes
2answers
145 views

Is a path connected subspace of a simply connected space simply connected?

This is sort of a lemma I'm trying to prove for a larger proof. It seems intuitively true: if a space has trivial fundamental group, any two loops based at a point are homotopic. A subspace of such a ...
1
vote
1answer
163 views

Continuous maps from $S^1 \to X$ equivalent conditions

The following are equivalent for a topological space X according to a problem in Hatcher. $1$)Every continuous map $S^1 \to X$ is homotopic to a constant map. $2$)Every continuous map $S^1 \to X$ ...
1
vote
1answer
57 views

2-connected map between a connected sum and a gluing along one point.

I'm working with closed 3-dimensional manifolds $M_1$ and $M_2$. Consider their connected sum $M_1\#M_2$ and their gluing at only one point $M_1\vee M_2$. Intuitively I think that the map that ...
1
vote
1answer
60 views

Showing homotopy of two paths if they are homotopic after a delay

Let $X$ be a topological space and let $\gamma, \delta : [0,1] \rightarrow X$ be two paths from $x$ to $y$. Now define $\widehat{\gamma}: [0,2] \rightarrow X$ by $$\widehat{\gamma}(t) = \begin{cases} ...
2
votes
0answers
217 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
7
votes
3answers
255 views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$? consider this quotient on the disk representing the situation: $\mathbb{RP}^2$ (sorry ...
1
vote
1answer
111 views

homotopies of paths in the cylinder

Well I tried to imagine all this paths in my mind ( actually I used a glass and wool) to define informally the homotopy between these four paths, I think that they are homotopic, but how can I prove ...
3
votes
0answers
102 views

The fundamental group of the union of three convex open subsets of $ \mathbb{R}^n$.

I have to prove that the fundamental group of the union of three open convex subsets of $\mathbb{R}^n$ is trivial or $\mathbb{Z}$. I can show that it has only one generator, but I can't prove that if ...
2
votes
0answers
70 views

The fundamental group of an open set in $\mathbb{R} ^n$ does not have nilpotent elements.

I am studying a little of basic algebraic topology and I thought that this statement could be true. If you have an open connected set $U \subset \mathbb{R}^m$ and a loop $\gamma$ that is not ...
1
vote
1answer
203 views

Prove that the spaces have the same homotopy type

This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima. "Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...