Volume 8, issue 1 (2004)

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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 $SO\left(3\right)$. 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
Publication
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