# A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

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?

• In the world of manifolds the h-cobordism theorem adresses such questions. – HenrikRüping Jun 26 at 10:56
• @HenrikRüping is it the case that homotopy equivalences necessarily give rise to h-cobordisms? – Connor Malin Jun 26 at 12:12
• 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 $P^2$. – Ben Wieland Jun 26 at 17:26