Re: Number of non-diffeomorphic smooth structures on S^7.

From: jfdavis (jfdavis_at_indiana.edu)
Date: 02/20/05 

Date: 20 Feb 2005 15:15:00 -0500

In article <cv4o0c$40r$1@dizzy.math.ohio-state.edu>,
 mathjlp@yahoo.com (L.P) wrote:

> After asking for a proof of the fact that there are 28
> non-diffeomorphic
> structures on the 7-sphere I have been refered to the article "Groups
> of homotopy
> spheres I" by Milnor and Kervaire from 1962 in Annals.
>
> They construct and study the group \teta_n of h-cobordism classes of
> homotopy
> n-spheres and show eg. that for n=7 this group has 28 elements.
>
> I have the following questions :
>
> 1) Does it follow immediately that the order of this group equals the
> number in the header ??
>
> 2) Did they ever write a second article ??
>
> Thanks in advance.

No, they never wrote a second article, but the article by J. Levine:

 Lectures on groups of homotopy spheres. Algebraic and geometric
topology (New Brunswick, N.J., 1983), 62--95, Lecture Notes in Math.,
1126, Springer, Berlin, 1985.

is a good substitute. The group of exotic (oriented) 7-spheres has
order 28 - where the group structure is connected sum. It is cyclic
since every exotic 7-sphere bounds a parallelizable manifold, and the
group bP_8 is detected by signature. I believe that any smooth
structure on a 7-sphere not diffeomorphic to S^7 is not parallelizable,
but I did not double check this.

Jim Davis