Date: Thu, 18 Jan 2001 15:14:12 -0500
From: Frank Connolly <connolly.1@nd.edu>
Subject: Re: embedding question

Concerning Bill Dwyer's question:
        The results of DeSapio (Annals, 1965) prove that a stably
parallelizable manifold N of dimension n (with n at least 5) embeds in
euclidean space of dimension 2n-1. There is one possible exception--when n
is congruent to 6 mod 8 and the Kervaire Invariant of N remains non-zero no
matter what framing is chosen.
(I suspect even this possible exception can be dispatched by some
geometrical expedient).
        Moreover if N is k-connected, DeSapio shows the euclidean space's
dimension can be improved to 2n-2k-1, with the same exception, but also k
must be smaller than [n/4].
        Of course, by Hirsch, N immerses in R^(n+1), and by Whitney, N
embeds in R^2n (on this point, I am confused by Mark Mahowald's comment).

        Actually this question from Bill (eventually getting to me), and my
response technique, must represent some kind of a record in mathematical
non-communication between colleagues: not only is Bill's office a few steps
from mine, but also we just finished arranging to eat dinner together
tonight!

                        Frank Connolly