#### 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.

Notes on file formats
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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