Module 3 : Fundamental groups & its basic properties
Lecture 11 : Homotopies of maps. Deformation retracts
 

Theorem 11.6:

If $ f : X \longrightarrow Y$ is a homotopy equivalence then the groups $ \pi_1(X, x_0)$ and $ \pi_1(Y, f(x_0))$ are isomorphic.

Proof:
There exists $ g: Y \longrightarrow X$ such that $ f\circ g$ and $ g\circ f$ are respectively homotopic to id$ _{Y}$ and id$ _{X}$. By theorem 11.5 $ f_*\circ g_*$ differs from the identity map on $ \pi_1(Y, (f\circ g)(y_0))$ by a composition with the isomorphism $ h_{[\sigma]}$ where $ \sigma$ is a path joining $ f(g(y_0))$ and $ y_0$. In particular $ f_*\circ g_*$ is bijective and so $ f_*$ is surjective and $ g_*$ is injective. Likewise, working with $ g\circ f$ one concludes that $ g_*$ is surjective and $ f_*$ is injective. Hence $ f_*$ is an isomorphism between $ \pi_1(X, x_0)$ and $ \pi_1(Y, f(x_0))$.