Prove isomorphism of fundamental groups

Question

Hei, guys! I'm having some problem solving the next exercise:

Let $f: M -> N$ be a homeomorphism. Define a map $f*:π_1 (M, x_0) → π_1 (N, f(x_0 ))$ such that $f*([\gamma])=[f∘\gamma]$. Show that $f*$ is an isomorphism. Check that the map is well-defined.

Any suggestions?

Answer 1

We just need to prove that the fundamental group satisfies two properties below.

• If $\text{id}:(M,x_0)\to(M,x_0)$, then $\text{id}_*=\text{id}:\pi(M,x_0)\to\pi(M,x_0)$.

• If $f:(L,x_0)\to(M,y_0)$, $g:(M,y_0)\to(N,z_0)$, then $(g\circ f)_*=g_*\circ f_*:\pi(L,x_0)\to\pi(M,z_0)$.

So if $f$, $g=f^{-1}$ are homeomorphism, then $$\text{id}=\text{id}_*=(f\circ g)_*=f_*\circ g_*$$ $$\text{id}=\text{id}_*=(g\circ f)_*=g_*\circ f_*$$ Now we get the conclusion.

Actually the proof is based on the theory of functor: http://en.wikipedia.org/wiki/Functor.

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If you want prove it by the homotopy, I shall give the next theorem.

Theorem Suppose $f$, $g:M\to N$ are continuous, and $H$ is the homotopy map between them. For any $q\in X$, let $h$ be the path in $Y$ from $f(q)$ to $g(q)$ defined by $h(t)=H(q,t)$. Now we have an isomorphism between $\pi(X,f(q))$ and $\pi(Y,g(q))$.

You can find clue in the Lee's book: Introduction to Topological Manifold, P164.