Showing that a map $f$ between connected $CW$-complexes is a homotopy equivalence.

Question

Showing that a map $f$ between connected $CW$-complexes is a homotopy equivalence if it induces an isomorphism on $\pi_{1}$ and if a lift $\tilde{f} : \tilde{X} \rightarrow \tilde{Y}$ to the universal covers induces an isomorphism in homology.

Could anyone give me a hint for the solution?

Answer 1

Relative hurewicz applied to $\tilde f$ shows that it is an isomorphism on homotopy groups, but $\pi_n \tilde X = \pi_n X$ for $n>1$, and this isomorphism commutes with the maps induced by $f$ and $\tilde f$.

So your map $f$ is an iso on all homotopy groups. Apply Whitehead.