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.)



Is every degree 1 self-map a homotopy equivalence?
I believe that this is an open question in general, and the assertion is an old conjecture of Hopf. Some special cases were considered by Jean-Claude Hausmann, Geometric Hopfian and non-Hopfian situations. Geometry and topology (Athens, Ga., 1985), 157–166, Lecture Notes in Pure and Appl. Math., 105, Dekker, New York, 1987. I don't think there's been much progress since then, at least not that I could find via Mathscinet or Google Scholar.


It is easy to see that $f$ induces a surjection on $\pi\_1$; if not, then $f$ factors through a non-trivial covering space of $M$, contradicting the degree-$1$ assumption. So if $\pi\_1$ is Hopfian (any surjection from G to G is an isomorphism) then you get that $f\_\*$ is an isomorphism on $\pi\_1$. There are, however, some non-Hopfian groups. Even when you know that $f\_\*$ is an isomorphism, you need more to show $f$ to be a homotopy equivalence; you'd need $f$ to induces homology isomorphisms with coefficients in $Z[\pi\_1]$.


