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I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this paper. Most of these numbers are less than $10$, and all of them except $15$ is less than a thousand. But then $15$ has $16,256$ different classes.

I always thought of higher dimensional spheres as being fairly homogeneous as you went out, so this variance came as a surprise. My question is whether there is an easy way to explain such a large jump. Also, is there a more current table of values available?

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    19 seems to be more special!http://en.wikipedia.org/wiki/Exotic_sphere2011-12-27
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    See the exotic $R^4$ as well, http://en.wikipedia.org/wiki/Exotic_R42011-12-27
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    I had assumed 15 wasn't really special. I would be surprised in fact if the jumps didn't get arbitrarily large. The jumps seem to happen at 4k-1 dimensions, so what's happening at those numbers?2011-12-27
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    @Matt, dimension $4k-1$ is indeed special. Heuristically speaking, exotic spheres are exotic either b/c they don't bound parallelizable manifolds or they bound parallelizable manifolds that aren't contractible. A manifold of dimension $4k$ has room to have very rich signature obstructions to being contractible.2012-03-23

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