Does there exist a connected topological manifold $M$ such that $M\{pt\}$ is nonsmoothable? My understanding is that Quinn showed that these are always smoothable in dimension 4 (in fact in uncountably many ways, due to Gompf). Googling "almost smooth", which seems to be the relevant terminology, didn't help.
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1$\begingroup$ I never thought seriously about nonsmoothable manifolds but here is a sketch which I suspect should work. Start from a nonsmoothable $4$manifold $M$. Then $M\times T^k$ is has no PL structure, where $T^k$ is the $k$torus. By MayerVietoris the inclusion $M\times T^k \{pt\}\to M\times T^k$ is onto on $H^4$, so the KirbySiebenmann obstruction to the existence of a PL structure survives the restriction to $M\times T^k \{pt\}$, which therefore has no PL structure. The references are in en.wikipedia.org/wiki/Kirby%E2%80%93Siebenmann_class. $\endgroup$ – Igor Belegradek Apr 16 '18 at 2:17

1$\begingroup$ In general, the tangent microbundle to a nonPL manifold N is not in the image of $[N, BPL]\to [N, BTOP]$ so it is mapped to a nonzero element under the next map in the homotopy sequence $[N, BTOP]\to [N, BTop/PL]\cong H^4(N,\mathbb Z_2)$. When you remove a point the tangent microbundle should restrict, so the the inclusioninduced map on 4th homology preserves the KirbySiebenmann class. Again, I am not posting it as an answer because I haven't though about this sufficiently but it seems correct. $\endgroup$ – Igor Belegradek Apr 16 '18 at 2:23

$\begingroup$ In fact what this argument seems to give is that if $N$ is a topological $n$manifold without a PL structure and $n>4$, then $N\{pt\}$ has no PL structure. $\endgroup$ – Igor Belegradek Apr 16 '18 at 2:39