Timeline for Are there non-smoothable homotopy/homology spheres?

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Oct 18, 2021 at 22:59	comment	added	horned-sphere		I think your first sentence might go at the heart of Albanese's question. More precisely, it seems that back in 1960-62 when Smale, Zeeman, and Stallings proved the Topological Poincaré Conjecture for $n\geq 5$ their proofs all assumed a homotopy sphere to be smoothable, or at least PL (in contrast with Albanese's definition). Hence if one goes off of those papers we don't know that any homotopy $n$-sphere is homeomorphic to $S^n$. That is to say, it wasn't until M.H.A.Newman in "The Engulfing Theorem for Topological Manifolds" ('66) that there was a proof of the full Poincaré for $n\geq 5$.
Apr 14, 2016 at 19:12	vote	accept	Michael Albanese
Apr 13, 2016 at 20:54	comment	added	Igor Belegradek		I do not know such examples, but then if you remember Freedman's work in the simply-connected case the non-smoothable examples arise in a roundabout way. It might be that the same will be true for 4-manifolds with large fundamental group, i.e. a smoothable manifold may have a non-smoothable "evil twin" which has the same intersection form and nontrivial Kirby–Siebenmann invariant.
Apr 13, 2016 at 20:23	history	edited	Michael Albanese	CC BY-SA 3.0	added 2 characters in body
Apr 13, 2016 at 16:45	comment	added	Michael Albanese		Are there any explicit examples of homology $4$-spheres for which it is not yet known if they admit a smooth structure or not?
Apr 13, 2016 at 1:22	history	answered	Igor Belegradek	CC BY-SA 3.0