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

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

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

• Why in the earth do you think that the 28 differentiable structures of $S^7$ have some relation with the Mersenne numbers? – Martín-Blas Pérez Pinilla Mar 2 '18 at 13:31
• I don't see how that comment could contribute to the discussion. Do the 28 structures have a connection with mersenne numbers, or don't they? And if they don't, why not just say so? – MJD Mar 2 '18 at 13:40

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.