In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this really true?
Using Poincare duality, it is easy to see that $f$ is a homology equivalence. But has $f$ to induce an isomorphism on $\pi_1$? Another (maybe related) result is Hopf's theorem: The degree classifies maps $M \rightarrow S^n$ up to homotopy equivalence.
(I am sorry if this question is too basic. Feel free to delete it in this case.)