Claim: if $h\colon(X,a)\to(Y,b)$ is a homeomorphism of $X$ with $Y$, then $h_*\colon \pi_1(X,a)\to \pi_1(Y,b)$ is an isomorphism.
where $\pi_1$ refers to the fundamental group and $h_*$ is the induced homomorphism defined by $h_*([f]) = [h(f)]$.
I already know $h_*$ is a homomorphism since $h(f\cdot g)=h(f) \cdot h(g)$. To show $h_*$ is an isomorphism, I thought it sufficed to show it's a bijection...
Munkres' Proof. Let $k: (Y,b)\to(X,a)$ be the inverse of $h$. Then $k_*\circ h_*=(k\circ h)_* = i_*$, where $i$ is the identity map of $(X,a)$. And $h_*\circ k_*= (h\circ k)_*=j_*$, where $j$ is the identity of $(Y,b)$. Since $i_*$ and $j_*$ are the identity homomorphisms of the groups $\pi_1(X,a)$ and $\pi_1(Y,b)$, respectively, $k_*$ is the inverse of $h_*$. $\Box$
How does this show $h_*$ is a bijection? Since $h$ is a homeomorphism, I know $h_*$ is injective.