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.)

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    $\begingroup$ It is in fact Problem 5.26 of Kirby's problem list (math.berkeley.edu/~kirby/problems.ps.gz), and still open as far as I know. $\endgroup$ – skupers Jan 23 '15 at 6:40
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    $\begingroup$ I don't know what dimension you're interested in, but perhaps it's worth pointing out that 3-manifold groups are known to be Hopfian. The upshot is that the assertion is true for irreducible 3-manifolds with infinite fundamental group, and perhaps also in greater generality than that... $\endgroup$ – HJRW Jan 27 '15 at 9:45

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 $Z[\pi_1]$.


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