Induced Map

If $f:(X,A) \to (Y,B)$ is homotopic to $g:(X,A) \to (Y,B)$, then $f_*:H_n(X,A)\to H_n(Y,B)$ and $g_*:H_n(X,A)\to H_n(Y,B)$ are said to be the induced maps.

See also Eilenberg-Steenrod Axioms