Date: Mon, 05 Feb 2001 13:44:42 +0100
From: Andrew Ranicki <aar@maths.ed.ac.uk>
Subject: Re: more on embedding question

Kervaire ("On higher dimensional knots", Morse Symposium, 1965) proved
that a homotopy n-sphere \Sigma^n can be smoothly embedded in S^{n+2}
if and only if \Sigma^n is the boundary of a parallelizable
(n+1)-manifold.

Andrew Ranicki
aar@maths.ed.ac.uk
____________________________
From: Mark Mahowald <mark@math.northwestern.edu>
Subject: Re: more on embedding question
Date: Mon, 5 Feb 2001 16:05:25 -0600 (CST)

I think Massey proved something like: Any exotic n-sphere embeds no
better than in R^{n+3}. A possible reference is in the Proc. AMS Vol
10,(1959)959-964.

A thesis student of mine "proved" that the exotic spheres associated
to the homotopy classes call eta_j in stem 2^j do not embed in
R^{2^j+2{j-1}-3}. I use quotes about proved since he never published
the result. I suppose this makes it at least a conjecture.

Mark Mahowald
_______________________________________
Date: Mon, 05 Feb 2001 17:10:16 -0500
From: "John R. Klein" <klein@math.wayne.edu>
Subject: Re: more on embedding question

> Another natural question that arises from Tom's answer is: If the exotic
> seven-spheres don't embed in R^8, then where *do* they embed?

1) They embed in codimension 3 (using the work of Haefliger).

2) They embed in codimension 2: This uses Th 1.7 of

Hsiang, W.-c.; Levine, J.; Szczarba, R. H.
On the normal bundle of a homotopy sphere embedded in Euclidean space.
Topology 3 1965 173--181.

and the work of Kervaire and Milnor that every exotic 7-sphere
is the boundary of an paralellizable 8-manifold.
_____________________________________________________
From: Dave Rusin <rusin@math.niu.edu>
Date: Mon, 5 Feb 2001 16:44:59 -0600 (CST)
Subject: Re:  more on embedding question

>Another natural question that arises from Tom's answer is: If the exotic
>seven-spheres don't embed in R^8, then where *do* they embed?

Ooh! Ooh! I know! I know that one!

Consider the complex hypersurface of C^5 defined by
z1^(4k+1) + z2^3 + z3^2 + z4^2 + z5^2 = 0.
There's a singularity at the origin but small spheres intersect the
surface transversely, giving a 7-manifold embedded in R^10. Take
k=0, 1, 2, ..., 27  to get all diffeomorphism classes of 7-spheres.

dave