Maps between projective planes inducing the isomorphism of fundamental groups

Question

Let $P^2$ be the projective plane and $f:P^2 \to P^2$ a continuous map inducing an isomorphism $f_*: \pi_1(P^2) \to \pi_1(P^2)$.

Can we prove that $f$ must be homotopic to a homeomorphism between $P^2$?

Answer 1

No.

Consider a map $f: S^1 \to S^1,$ $f: z \mapsto z^3.$ Then $\Sigma f: S^2 \to S^2$ induces a map $\pi_2(S^2) \xrightarrow{\times3} \pi_2(S^2).$ Moreover, since $(-z,-t) \mapsto (-z^3,-t) = -(z^3,t),$ $\Sigma f$ descends to a map $P^2 \to P^2,$ and since the universal covering $S^2 \to P^2$ induces an isomorphism in $\pi_2,$ this map will not be homotopic to a homeomorphism. But on $\pi_1(P^2)$ it takes the generator to thrice the generator, so is an insomorphism.