I was try to understand the following theorem:-
Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic.
Proof: The fundamental groups of both the spaces $X$ and $Y$ are independent on the base points since they are path connected. Since $X$ and $Y$ are of the same homotopy type, there exist continuous maps $f:X\to Y $ and $g:Y\to X$ such that $g\circ f\sim I_X$ by a homotopy, say, $F$ and $f\circ g \sim I_Y$ by some homotopy, say $G$. Let $x_0\in X$ be a base point. Let
$$f_\#:\pi_1(X,x_0)\to \pi_1(Y,f(x_0))$$ and $$g_\#:\pi_1(Y,f(x_0))\to \pi_1(X,g(f(x_0)))$$ be the induced homomorphisms. Let $\sigma$ be the path joining $x_0$ to $gf(x_0)$ defined by the homotopy $F$.
After that the author says that $\sigma_\#$ is a isomorphism. obviously $\sigma_\#$ is a homomorphism but I could not understand how it becomes a isomorphism.
Can someone explain me please. thanks for your kind help and time.
Best Answer
In my humble opinion, your proof is too much ad hoc. Just show that $f,f' : (X,x) \to (Y,y)$ induced the same homomorphism of groups $\pi_1(X,x) \to \pi_1(Y,y)$ when they are homotopic : this is sufficient, the rest will come from abstract non sense.