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If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?

It is not true, for instance, that if there is an orientation-reversing homeomorphism there must be an orientation-reversing diffeomorphism (consider exotic 7-spheres).

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    $\begingroup$ According to arxiv.org/pdf/0907.5283.pdf (Section 2, 4th bullet point), the lens space $L_5(1,1)$ has an orientation reversing homotopy equivalence, but no orientation reversing homeomorphism. Of course, it is not simply connected. $\endgroup$
    – Jason DeVito - on hiatus
    Commented Oct 5, 2020 at 16:37
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    $\begingroup$ The difference between homeomorphism and diffeomorphism is subtle (torsion). The difference between homotopy equivalence and homeomorphism is crude (rational), the same as the difference between homotopy equivalence and diffeomorphism. Examples involving exotic spheres are difficult and overkill. There are easy smooth examples using Pontrjagin classes. There are linear $S^4$ bundles over $S^4$ homotopy equivalent to $S^4\times S^4$ with nonzero $p_2$. Obviously smoothly chiral. Given the difficult but blackbox theorem that $p_2$ is a rational homeomorphism invariant, topologically chiral. $\endgroup$
    – Ben Wieland
    Commented Oct 6, 2020 at 19:07
  • $\begingroup$ @BenWieland Do you know of a standard reference for the obstructions to homotoping a homotopy equivalence to a homemorphism? And for isotopy between a homeomorphism and a diffeomorphism? $\endgroup$
    – Connor Malin
    Commented Oct 7, 2020 at 2:23
  • $\begingroup$ @ConnorMalin I'm not really sure what you're asking, but the first topic is called surgery theory and the second topic is called smoothing theory (or triangulation theory). For most purposes, I recommend the two volumes "Surveys on Surgery Theory," ed Cappell-Ranicki-Rosenberg. But if you really want a foundational reference for the topological category, I think the only option is Kirby-Siebenmann. $\endgroup$
    – Ben Wieland
    Commented Oct 8, 2020 at 19:30
  • $\begingroup$ Correction: there is a smooth manifold homotopy equivalent to $S^4\times S^4$ with nonzero $p_2$ and thus no orientation reversing homeomorphism, but I don't see an easy way to construct it. However, linear sphere bundles have nontrivial $p_1$ and so do provide counterexamples to the more natural question of whether a particular homotopy equivalence is homotopic to a homeomorphism. That is, the homotopy equivalence which reverses orientation on the base sphere is not homotopic to a homeomorphism, even though the homotopy equivalence which reverses orientation on fiber is realized by a homeo. $\endgroup$
    – Ben Wieland
    Commented Oct 9, 2020 at 13:46

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