Date: Wed, 24 Jan 2001 12:47:40 -0500 (EST)
From: taylor.2@nd.edu
Subject: Re: embeddings

On the embedibility of M^4 in R^7 Browder asks
What about non-smooth manifolds?
Does Donaldson cover the non-simply connected case?

Non-simply connected manifolds are covered by Donaldson
as outlined by Connolly. Donaldson's diagonalization
theorem does not need simply connected. The characteristic
element on H^2/tor that Connolly writes down,
V= x_1 + x_2 + ... + x_n, may not reduce
to w_2, but there is a 2-torsion element T such that
V+T reduces to w_2 and the square is unchanged.

Non-smooth is more fun.
Fang Fuquan (Topology 1994) purports to prove that
the Boechat-Haefliger result holds for locally-flat
embeddings, but the proof is wrong. It seems likely
that the Boechat-Haefliger result does hold for 4-manifolds
which embed in R^7 with a smooth normal bundle and that
Fang Fuquan has proved this.

In the other direction, one can see from Freedman that all
simply-connected 4-manifolds have locally-flat embeddings
in R^7 and I suspect all orientable 4-manifolds have such
embeddings.

Larry Taylor