Volume 8, issue 1 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Witten's conjecture and Property P

Peter B Kronheimer and Tomasz S Mrowka

Geometry & Topology 8 (2004) 295–310

arXiv: math.GT/0311489

Abstract

Let K be a non-trivial knot in the 3–sphere and let Y be the 3–manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a non-trivial homomorphism to the group SO(3). In particular, Y cannot be a homotopy-sphere.

Keywords
3–manifold, knot, surgery, homotopy sphere, gauge theory
Mathematical Subject Classification 2000
Primary: 57M25, 57R57
Secondary: 57R17
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Publication
Received: 7 December 2003
Revised: 9 December 2003
Accepted: 13 February 2004
Published: 14 February 2004
Proposed: Robion Kirby
Seconded: John Morgan, Ronald Stern
Authors
Peter B Kronheimer
Department of Mathematics
Harvard University
Cambridge
Massachusetts 02138
USA
Tomasz S Mrowka
Department of Mathematics
Massachusetts Institute of Technology
Cambridge
Massachusetts 02139
USA