Let K be a
knot in S3, and let
Yp∕q(K) be the 3-manifold obtained
by Dehn surgery on K
with surgery coefficient p∕q∈ℚ∪∞.
In the case p=1,
Y1∕q(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),
Y1∕q(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(Y1∕q) has a non-trivial
homomorphic image in SU(2),
for all q≠0.