Every homomorphism on cohomology groups is induced by some map on spaces

Question

Suppose we have two topological spaces $X$ and $Y$. Let $\alpha ^* :H^n(X,\mathbb{Z})\to H^n(Y,\mathbb{Z})$ be a homomorphism.

1) Is there a continuous map $\alpha : Y\to X$ such that the induced map $H^n(\alpha)$ equals $\alpha ^*$?

2) What if we replace cohomology with homology?

Answer 1

Not necessarily, in both cases. Take as an example $X=S^2, Y=T^2$. $\pi_2(T^2) = 0$ so any map $S^2\to T^2$ is nullhomotopic, in particular it induces the $0$ map on (co)homology.

However $H^2(S^2) \cong H^2(T^2) \cong \mathbb Z$ (and same in homology)