From: ruberman@brandeis.edu (Daniel Ruberman)
Subject: Re: Embeddings
Date: 15 Jun 1999 09:00:02 -0500
Newsgroups: sci.math.research
Keywords: Do knot complements determine knot type? 

In article <7k40tt$m3d$1@its.hooked.net>, daz@well.com (Daz) wrote:

> In article <7jv8n1$pjo@sjx-ixn6.ix.netcom.com>,
> Daniel Giaimo <dgiaimo@ix.netcom.com> wrote:
> >    Suppose I have two embeddings of a compact manifold in R^n.  Then is it
> >true that they are homotopically equivalent iff their complements are
> >homeomorphic?
>
-------------------------------------------------------------------------------
> 
> I'm not exactly sure what "homotopically equivalent" means here.   If it just
> means "homotopic as maps", then trivially all maps into a contractible space
> like R^n are homotopic.   (And examples abound of embeddings of a given
> manifold into R^n having non-homeomorphic complements.)
> 
> There is an old question in knot theory about whether two embeddings f,g of 
> a manifold M into R^n having homeomorphic complements are necessarily 
> the same knot (i.e. are the pairs (R^n,f(M)) and (R^n,g(M)) homeomorphic).
> This was shown some time ago for knotted spheres S^n in R^(n+2) for n > 2, 
> and more recently for conventional knots S^1 in R^3, but is false for
> links in R^3.
> 
> Dan Asimov
> asimov@msri.org

The second paragraph is not correct: There are examples of
Cappell-Shaneson (Annals 1976) of knots in dimensions 4,5,6,7, and Suciu
(Commentarii 1992) (in dimensions congruent to 3 or 4 mod 8) of
inequivalent knots with the homeomorphic complements (also Gordon,
Commentarii 1976, for dimension 4).  The classical theorem says that there
are at most two knotted spheres with the same complements.

One interpretation of the original question is whether knots with homotopy
equivalent complements have homeomorphic complements.  The answer to this
is also no, even in the classical dimension: The granny and square knots
(each connected sums of trefoils) have homotopy equivalent complements,
but no homotopy equivalence takes the boundary to the boundary.  Examples
in higher dimensions were first given in Cappell's thesis (in the Georgia
Topology conference proceedings, 1968).

Another interpretation of the original question is whether two
(potentially different) manifolds are homotopy equivalent iff they have
embeddings into some R^n with homeomorphic complements.  The answer is
almost certainly no, but I don't know an example off the top of my head.

Daniel Ruberman