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Let $M^{3}$ be a closed orientable 3-manifold, and $\phi:H_{1}(M;\mathbb{Z})\to H_{1}(M;\mathbb{Z})$ be an automorphism of abelian groups.

My question is: Is there any characterization of $\phi$ ensuring that $\phi$ is induced by some self homotopy equivalence of $M$?

Actually, I'm mainly interested in the case when $M$ is hyperbolic. One may also replace "homotopy equivalence" by "homeomorphism".

Thanks!

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  • $\begingroup$ Do you mean self-homotopy equivalences? $\endgroup$
    – Alex Degtyarev
    Commented May 23, 2015 at 7:38
  • $\begingroup$ @AlexDegtyarev Yes. Sorry for carelessness. $\endgroup$
    – Haimiao Chen
    Commented May 23, 2015 at 7:40
  • $\begingroup$ I don't know anything about hyperbolic 3-manifolds, but there is a general obstruction theory for lifting maps in (co)homology to actual homotopy classes of maps. The first obstruction is that the map on cohomology be a map of modules over the Steenrod algebra, but I don't think that's much of a condition in your case since 3 is a small number. After that there is a sequence of obstructions living in algebraically defined Andre-Quillen style groups in the category of unstable modules over the Steenrod algebra. That sounds very complicated and it is, but in this case there's not much to compute $\endgroup$
    – Dylan Wilson
    Commented May 23, 2015 at 9:21
  • $\begingroup$ (cont'd) so presumably someone who knows more about these manifolds could compute them. That gives you something up to p-completion, so you'll have to do something rationally as well and then glue. See, e.g., Goerss-Jardine VIII.4 for this obstruction theory spelled out, and also the work of Lannes etc. $\endgroup$
    – Dylan Wilson
    Commented May 23, 2015 at 9:23
  • $\begingroup$ @Dylan: I think here all you need to know about hyperbolic manifolds $M$ is that they're aspherical. So, writing $G = \pi_1(M)$, the question is equivalent to: when does an automorphism $G/[G, G] \to G/[G, G]$ lift to an automorphism $G \to G$? I'd expect the obstruction theory to be simpler because of this, but maybe not. $\endgroup$
    – Qiaochu Yuan
    Commented May 24, 2015 at 4:06

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Since you expressed interest in the hyperbolic case: it follows from the Mostow rigidity theorem and the Hurewicz theorem that an isomorphism $\phi \colon H_1(M) \to H_1(M)$ is induced by an isometry if $\phi$ is the abelianization of an isomorphism $\psi \colon \pi_1(M) \to \pi_1(M)$. This statement applies to any hyperbolic manifold of dimension 3 or higher. I doubt it gives a complete characterization, but maybe it helps.

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    $\begingroup$ Actually the main difficulty lies in deciding whether $\phi$ is induced by some $\psi:\pi_{1}(M)\to\pi_{1}(M)$. $\endgroup$
    – Haimiao Chen
    Commented May 23, 2015 at 23:31

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