Does there exist a connected topological manifold $M$ such that $M-\{pt\}$ is non-smoothable? 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 non-smoothable manifolds but here is a sketch which I suspect should work. Start from a non-smoothable $4$-manifold $M$. Then $M\times T^k$ is has no PL structure, where $T^k$ is the $k$-torus. By Mayer-Vietoris the inclusion $M\times T^k -\{pt\}\to M\times T^k$ is onto on $H^4$, so the Kirby-Siebenmann 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 BelegradekApr 16, 2018 at 2:17
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2$\begingroup$ In general, the tangent microbundle to a non-PL 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 inclusion-induced map on 4th homology preserves the Kirby-Siebenmann class. Again, I am not posting it as an answer because I haven't though about this sufficiently but it seems correct. $\endgroup$– Igor BelegradekApr 16, 2018 at 2:23
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$\begingroup$ 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 BelegradekApr 16, 2018 at 2:39
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