From: danield@mindless.com
Date: Mon, 21 Feb 2000 15:26:00 -0500 (EST)
Subject: Some questions regarding Milnor Spheres

Hello,

My name is Daniel Alayón-Solarz, I am a pregraduate math student at Universidad
Simón Bolívar, in Venezuela.
I am interested in Milnor 7-Spheres which are  homeomorphic but not diffeomorphi
c to the standard 7-sphere.

The first thing I noticed is that if a single point is removed from a Exotic Sph
ere the resulting manifold is homeomorphic and diffeomorphic to R^7, hence Stand
ard and Exotic Spheres are diffeomorphic up to a single point (this is done via
a Morse function with only two critic non-degenerate points). Suppose you start
with  a Exotic Sphere, then you remove a  single point and get R^7, is there any
 way to reverse the process? i.e How to Add a Point (at infinity) to R^7 in orde
r to obtain a non-Standard Sphere? I think this could be done if you had an expl
icit homeomorphism (such that it is a diffeomorphism up to a single point) betwe
en a Standard and a Exotic Sphere (such a explicit homeomorphism has to exists)
in order to get an "exotic stereographic projection". I may be wrong about this
so it would be nice to know wether somebody can help me about it.
The second question is about finding a "differential invariant" rather than a to
pological one (which does not disting between a Exotic and a Standard Sphere), c
an this be done in any way?
The third and last one is pretty like the second one, Is there any smooth field
on a Exotic Sphere such that it is not Smooth in the standard one?
It is important to remember that any sphere can be realized as  2 disc whose bor
der (a (n-1)-Sphere) are identified by a diffeomorphism g:S^(n-1) -> S^(n-1), in
 the case of a Exotic Sphere the diffeomorphism g: S^6 -> S^6 (of degree +1) is
not smoothly isotopic to the identity, so what about finding an invariant under
isotopy

I would thank any kind of help about this subject.

Daniel D. Alayón-Solarz

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