Is there a $31$-dimensional manifold with 496 differential structures?

Question

Milnor found a $7$-dimensional sphere with 28 differential structures.

Is there a $31$-dimensional manifold with 496 differential structures?

Answer 1

It's "the sphere", or "any sphere", not "a sphere". Most likely you observed that 7 is a Mersenne prime and 28 is the associated perfect number. There's indeed a connection, though maybe not exactly what you expect: the number of smooth structures on the $(4k-1)$-sphere is divisible by $2^{2k-2}(2^{2k-1}-1)$; see wikipedia. When $k=2$ you happen to have $4k-1=2^{2k-1}-1=7$. On the 31-dimensional sphere ($k=8$), there are $7767211311104=4\cdot3617\cdot2^{2k-2}(2^{2k-1}-1)$ smooth structures, where 3617 is the numerator of $|4B_{16}/8|=-3617/1020$. On the other hand, there are $992=2\cdot 496$ smooth structures on the 11-sphere.

For smooth structures on Brieskorn manifolds (which Milnor's construction uses), you may want to look at https://mathoverflow.net/questions/126807/when-are-brieskorn-manifolds-homeomorphic