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 $N$?

6$\begingroup$ Worth pointing out that $M\times S^k$ is not smoothable for any $k$; see the lemma on page 219 of The Wild World of 4Manifolds by Scorpan. $\endgroup$ – Michael Albanese Sep 29 at 15:53

$\begingroup$ @MichaelAlbanese Shouldn't its product with $S^1$ be smoothable? The microbundle of $M$ is stably parallelizable, so the tangent bundle of this product should be trivial, correct? Then smoothing theory shows a parallelizable manifold has at least one smooth structure. $\endgroup$ – Connor Malin Sep 29 at 17:35

$\begingroup$ @ConnorMalin: I am not that familiar with microbundles, but wouldn't there be a nonzero $p_1$? $\endgroup$ – Michael Albanese Sep 29 at 18:00

$\begingroup$ @MichaelAlbanese I definitely could be misunderstanding, but I thought the Milnor manifolds (of which M is the first example) were constructed to show that the surgery obstruction map from the normal invariants of the sphere to $8\mathbb{Z}$ were surjective. Here the surgery obstruction map takes a normal invariant of the sphere to its signature. $\endgroup$ – Connor Malin Sep 29 at 18:10

$\begingroup$ To be the domain of a normal invariant of the sphere is equivalent to have your Spivak normal bundle be trivial, which is weaker than having the microbundle being stably parallelizable (which for a smoothable manifold means the tangent bundle is trivial as an $\mathbb{R}^n$ bundle), so perhaps I have misunderstood and in fact only the Spivak normal bundle is trivial, not the tangent microbundle. $\endgroup$ – Connor Malin Sep 29 at 18:12
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 KirbySiebenmann 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.

$\begingroup$ I should've noticed this. Thanks for pointing it out. $\endgroup$ – Michael Albanese Sep 29 at 17:58