Professor Peter Shalen
(University of Illinois at Chicago)
"Surfaces in knot exteriors, and 3-manifolds with cyclic fundamental group"
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.