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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!en.wikipedia.org/wiki/Exotic_sphere –  Ehsan M. Kermani Dec 27 '11 at 22:05
    
See the exotic $R^4$ as well, en.wikipedia.org/wiki/Exotic_R4 –  Ehsan M. Kermani Dec 27 '11 at 22:07
    
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? –  Joe Dec 27 '11 at 22:11
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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. –  Sam Lisi Mar 23 '12 at 23:36

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