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RESEARCH BLOG 10/17/03 Kronheimer, Mrowka, Ozsvath, and Szabo posted a paper on Mon-
 

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 non-trivial 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 write-up
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 4-manifold topologists are interested in surgery
on 3-manifolds. A smooth 4-manifold has a handle decomposition,
coming for example from a Morse function, with handles of five pos-

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics