KUWAIT FOUNDATION LECTURE 43 - October 26, 2004

DEHN SURGERY AND THE FUNDAMENTAL GROUP

Peter Kronheimer
(Harvard University)

Let K be a knot in S3, and let Y pq(K) be the 3-manifold obtained by Dehn surgery on K with surgery coefficient pq . In the case p = 1, Y 1q(K) has the homology of S3, and Bing (1963) asked whether such a manifold could be a counterexample to the Poincaré conjecture: simply connected, but not S3. According to Culler, Gordon, Luecke and Shalen (1987), Y 1q(K) cannot be simply connected if q > 1; and according to Gordon and Luecke (1989), Y ±1(K) cannot be S3 - both of these under the assumption that K is non-trivial. The lecture described recent work with Tom Mrowka, drawing on several recent strands in gauge theory and symplectic topology, which provides an answer to Bing’s question: if K is non-trivial, then π1(Y 1q) has a non-trivial homomorphic image in SU(2), for all q0.


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