From: William Browder <browder@imf.au.dk>
Subject: Re: 2 on embeddings
Date: Wed, 24 Jan 2001 8:47:28 MET

What about non-smooth manifolds?
Does Donaldson cover the non-simply connected case?

> _______________________________
>
> Date: Tue, 23 Jan 2001 16:48:34 -0500
> From: Frank Connolly <connolly.1@nd.edu>
> Subject: More on embeddings
>
>         EVERY COMPACT ORIENTABLE n-MANIFOLD EMBEDS IN S^{2n-1}.
>
> As Mark Mahowald explained earlier, Massey-Peterson (1964) proved, based on
> work of Haefliger-Hirsch, that if n is different from 4, an orientable
> n-manifold embeds in R^{2n-1}.
>
>         But it turns out that  if n=4, every smooth compact oriented 4
> manifold  embeds in R^7.
>
>          To see this, note that the paper of Boechat-Haefliger in the
> volume dedicated to deRham (Essays on Topology and Related Topics,
> Springer, 1970;  p.165), proves that a connected oriented compact
> 4-manifold M smoothly embeds in R^7 if and only if there is a
> CHARACTERISTIC class V in H^2(M;Z) (i.e. one whose reduction mod 2 is w_2)
> whose square, V^2, in H^4(M;Z)=Z, is equal to the index of M.
>         In 1970, they could not settle whether such a V could always be found.
>         However,  when the intersection form of M is indefinite, it is well
> known that this form contains a hyperbolic plane. Quite easy algebra using
> this hyperbolic plane then shows that any characteristic class V can be
> modified to get another such that V^2 = \tau.
>         When the intersection form is definite, the answer depends on
> Donaldson's 1987 theorem that the intersection form must be diagonal. If M
> is oriented to be positive definite, and x_1, x_2, ... x_n is an
> orthonormal basis, then V= x_1 + x_2 + ... + x_n  is the  class which
> Boechat-Haefliger require.
>
>         This is discussed briefly in a paper by Fang Fuquan in Topology 1994.