Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?

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    $\begingroup$ In the world of manifolds the h-cobordism theorem adresses such questions. $\endgroup$ – HenrikRüping Jun 26 at 10:56
  • $\begingroup$ @HenrikRüping is it the case that homotopy equivalences necessarily give rise to h-cobordisms? $\endgroup$ – Connor Malin Jun 26 at 12:12
  • $\begingroup$ This is an excessively specific question. A better question is whether there is a homotopy equivalence $X\to Y$ between distinct varieties that is not homotopic to a homeomorphism. If you have such an example, switching the factors on $V=X\times Y$ is a candidate answer to your question. Moreover, if you understand this new question, you can engineer a candidate that works. I believe that you can choose $X$ and $Y$ to be $P^2$ bundles over $P^2$. $\endgroup$ – Ben Wieland Jun 26 at 17:26

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