# Counting smooth structures on manifolds

Kervaire and Milnor found a formula for the number of smooth structures on the $4n - 1$ sphere (see, e.g. the last part of this MO answer). It is relatively easy to compute the number of smooth structures on the $k$ sphere for other values of $k$ (aside from $k = 4$).

Has there been work finding formulas for the number of smooth structures on elements of other infinite classes of manifolds?

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If it's so easy to count smooth structures on spheres, could you let me know how many there are on $S^{1020828302800}$? By and large I don't think there are people working on cute formulas for these things but there is a wide conceptual framework for the issue of relating smooth structures on manifolds to other things, it goes by the name of surgery theory. You can use it to count if you want, but counting for its own sake, I don't think people do that. – Ryan Budney Mar 6 '13 at 7:53
The set of smooth structures on $S^k$ forms a group, under connect sum, and this group acts on the set of smooth structures on $M^k$. I was under the impression that for $M$ compact connected smoothable, this action was simply transitive, but I don't know any references. – Allen Knutson Mar 6 '13 at 12:49
@Allen: in general the action is neither transitive, nor simply transitive, even in the simply-connected case. I do not know any single place where this is discussed, but some examples/discussion can be found in section 3 of arxiv.org/abs/0912.4874 which was written precisely to survey some methods on classifying closed smooth simply-connected manifolds. – Igor Belegradek Mar 6 '13 at 14:43
Googling "inertia group topology", I found this paper by Levine on the topic: maths.ed.ac.uk/~aar/papers/levineinert.pdf – Marco Golla Mar 6 '13 at 14:51
Incidentally, there is some demand for such classifications in Riemannian geometry; when geometers find/build new examples, say of nonnegative curvature or with special holonomy, they often want to be sure the examples are non-diffeomorphic to known ones. – Igor Belegradek Mar 6 '13 at 17:27