\begin{definition}[Definition:Induced Homomorphism Between Fundamental Groups]
Let $X,Y$ be topological spaces.
Let $f:X\to Y$ be a continuous map.
Let $x_0\in X$ and $y_0=f(x_0)\in Y$.
Let $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ be their fundamental groups.
The '''homomorphism induced by $f$''' is the group homomorphism $f_* : \pi_1(X,x_0) \to \pi_1(Y,y_0)$ defined by:
:$f_*([\gamma]) = [f\circ\gamma]$
\end{definition}