 
Summary: RESEARCH BLOG 10/17/03
Kronheimer, Mrowka, Ozsvath, and Szabo posted a paper on Mon
day, which proves many results about Dehn surgeries on knots (al
though the results depend on a paper of Kronheimer and Mrowka which
is in preparation). In particular, it gives a new proof of the knot com
plement problem, which is independent of Gordan and Luecke's proof.
The paper draws on many deep results and techniques of 3 and 4
manifold topology from the past couple of decades. When I was a
graduate student, I had heard from Mike Freedman that Kronheimer
and Mrowka were putting the finishing touches on a proof of property
P (which is the conjecture that no nontrivial Dehn filling on a knot
complement in S3
can give a homotopy sphere, which would be a con
sequence of PoincarŽe conjecture). I believe this paper is the writeup
of that proof, although they do not get homotopy information, possi
bly because they haven't proven homotopy invariance of the monopole
invariants.
Let's investigate why 4manifold topologists are interested in surgery
on 3manifolds. A smooth 4manifold has a handle decomposition,
coming for example from a Morse function, with handles of five pos
