Induced map on cohomology ($f^{\ast }$)

A continuous map $f:X\to Y$ induces a pullback $f^{\ast }:H^q(Y;R)\to H^q(X;R)$. It preserves sums and cup products, and $(g\circ f{)}^{\ast }=f^{\ast }\circ g^{\ast }$.

Homotopic continuous maps induce the same map on cohomology. In particular, a homotopy equivalence induces an isomorphism of cohomology rings.