Definition:Induced Homomorphism Between Fundamental Groups

Definition

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:

$f_*([\gamma]) = [f\circ\gamma]$

Also see

• Continuous Map Induces Group Homomorphism Between Fundamental Groups
• Definition:Induced Homomorphism Between Homotopy Groups

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $9$: The Fundamental Group $\S 52$: The Fundamental Group