Let $f:T^2 \to S^2$ be the map that collapses the 1-skeleton $S^1 \vee S^1$ Compute $f_∗$,$f^*$.
So start with the usual CW strucutre on the sphere (1 0-cell and 1 2-cell) and the torus (1 0-cell, 2 1-cells and 1 2-cell). Now all the boundary maps are zero in the chain complex of the torus. So let us map the 1-skeleton onto the zero cell and map the 2 cell of the torus onto the 2 cell of $S^n$ by the identity map. Since the boundary maps on the torus are 0, we have a continuous $f:T^2 \to S^2$ (I think?).
$f$ is the identity (edit: on the 2-cell), so does this imply that $f^*:H^2(S^2) \to H^2(T^2)$ and $f_*:H_2(T^2) \to H_2(S^2)$ are isomorphisms?
So I guess my answer to the question would be - the maps $f_n$ and $f^n$ are isomorphism's for $n=2$ and zero else?
Edit: Acutally I still have the 0 dimension to worry about as well I guess?