Timeline for Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$

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May 1, 2020 at 10:59	vote	accept	Kafka91
Apr 28, 2020 at 21:46	comment	added	Kevin Casto		It's probably worth mentioning Mostow rigidity (hyperbolic manifolds are smoothly rigid, and in fact hot-equiv implies isometric, for dim $\ge 3$) and the Borel conjecture (that aspherical manifolds are topologically rigid).
Apr 28, 2020 at 20:32	answer	added	Moishe Kohan		timeline score: 3
Apr 28, 2020 at 18:22	comment	added	Igor Belegradek		One approach to the diffeomorphism classification of closed manifolds in a given homotopy type is to compute the action of the monoid of homotopy self-equivalences on the structure set. You can find a discussion of these matters in arxiv.org/abs/0912.4874. For example in section 10 you can find a homotopy self-equivalence of $S^7\times CP^3$ with nontrivial invariant. In Remark 5.4 we discuss a self-homotopy equivalence with trivial normal invariant that is not homotopic to a diffeomorphism.
Apr 28, 2020 at 17:39	comment	added	Moishe Kohan		You are effectively asking for a smooth manifold $M$ such that there exists a self-homotopy-equivalence $M\to M$ which is not homotopic to a diffeomorphism. If one allows for noncompact $M$, then such examples exist already among surfaces (say, the triply-punctured 2-sphere). If you want compact examples, see my answer here.
Apr 28, 2020 at 13:50	history	edited	Kafka91	CC BY-SA 4.0	edited title
Apr 28, 2020 at 13:14	history	asked	Kafka91	CC BY-SA 4.0