Date: Thu, 18 Jan 2001 21:16:54 -0600
From: Bill Richter <richter@math.northwestern.edu>
Subject: Re: response on embeddings

   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).

Frank, I think Mark's right, and it might be explained in his paper

On obstruction theory in orientable fiber bundles. Trans. Amer.
Math. Soc. 110 1964 315--349

or maybe his paper with Paul Goerss

Immersions not regularly homotopic to embeddings.
Amer. J. Math. 109 (1987), no. 6, 1171--1195.

Trying to extend your `record in mathematical non-communication
between colleagues', I haven't looked at these paper recently.  Maybe
John Klein will :) Mark's certainly extended Whitney's embedding work,
and he may be blurring the exact line between Whitney and himself.

We can certainly say this: The top cell of your

    stably parallelizable manifold N of dimension n

falls off, so there's a stable degree 1 map S^n ---> N, which by
Freudenthal we can obtain unstably (not uniquely) in

pi_{2n-1} Sigma^{n-1} N

So N Poincare embeds in R^2n-1 by Bruce Williams's metastable Poincare
embedding theorem, since we're in a pi-pi situation.  (In my homotopy
theoretic proof in Duke J, the pi_1 don't cause any problems, nor does
it in Bruce's proof.)

Now Levine's proved a metastable smooth embedding theorem, but only in
the 1-connected case, where we already know there's an embedding, by
the Stallings embedding theorem, but even in the 1-connected case
there are surgery issues that you must understandd much better than I.
________________________________________
From: William Browder <browder@imf.au.dk>
Subject: Re: response on embeddings
Date: Fri, 19 Jan 2001 8:55:42 MET

Since I proved that the Kervaire invariant of such a manifold is 0
except possibly when n = 2^n - 2, that reduces the popssible exceptions.  Bill


> 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
>
>
>
>