The fundamental group of a loop space of any base point space $(X,x_0)$ is abelian?

Question

I was asked to prove that the fundamental group of a loop space of any base point space $(X, x_0)$ is abelian.

I thought that not every fundamental group of a base point space is abelian (for example, $S\vee S$, with the point that joins the two circles as the base point). Is the fundamental group of "loop space of a base point space $(X,x_0)$" not the same as the fundamental group of the base point space $(X,x_0)$? What am I missing?

Answer 1

This follow from the fact that$\pi_1(LM)=\pi_2(M)$ from the long exact sequence associated to the fibration $ev_x:PM\rightarrow M$ whose fibre is $LM$, apply the Serre exact sequence and the fact that $\pi_2(M)$ is commutative.

Here $PM$ is the space of continuous map $c:[0,1]\rightarrow M, c(0)=x$ and $ev_x(c)=c(1)$. $PM$ is contractible, to see this consider $H_u(c)(t)=c(tu)$, $H_1$ is the identity and $H_0(c)$ is the constant path $c(t)=x$.

Answer 2

Are you familiar with basic facts about homotopy groups? The higher homotopy groups are all abelian. As well, there is an isomorphism between $[S^n , X]$ and $[S^{n-1},\Omega X]$ where the brackets denote based homotopy classes. This isomorphism is an example of the adjunction between reduced suspension and loop spaces that exists on the level of continuous maps and homotopy classes of maps.

These facts combined tell you that the fundamental group is abelian, though this proof does not work for a general H-space.