Let $M$ be the $E\_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?



Smooth structure on direct product
Extending Michael Albanese's answer above, $M \times N^k$ will never be smoothable. For if it were then choose a point $p\in N$ and a chart U around $p$. Then $M \times U$ is an open subset of $M \times N$, and hence is smoothable. But as argued in Scorpan (p. 219, Lemma), $M \times \mathbb{R}^k$ is not smoothable.


Scorpan's argument uses the Kirby-Siebenmann product structure theorem to get down to dimension $5$. But I believe (it's been a while) that you could construct an argument based on Novikov's work on signatures.


