Geometry & Topology, Vol. 8 (2004) Paper no. 7, pages 295--310.

Witten's conjecture and Property P

P B Kronheimer and T S Mrowka

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

AMS subject classification. Primary: 57M25, 57R57. Secondary: 57R17.

DOI: 10.2140/gt.2004.8.295

E-print: arXiv:math.GT/0311489

Submitted to GT on 7 December 2003. (Revised 9 December 2003.) Paper accepted 13 February 2004. Paper published 14 February 2004.

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P B Kronheimer and T S Mrowka
Department of Mathematics, Harvard University
Cambridge MA 02138, USA
and
Department of Mathematics, Massachusetts Institute of Technology
Cambridge MA 02139, USA

Email: kronheim@math.harvard.edu, mrowka@math.mit.edu

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