Abstract:

Let M denote the exterior of a knot K in a closed 3-manifold Sigma. Using the variety of SL_2(C) representations of pi_1(M), Culler and I obtained necessary conditions involving the set of essential surfaces in M for Sigma to be simply connected, or more generally to have a cyclic fundamental group. This has given rise to an approach to the Poincar\'e Conjecture, on which progress is being made in joint work by Culler, N. Dunfield, W. Jaco and myself. I will a little about the ideas that go into this project in order to communicate something of the flavor of it.