Peter Kronheimer

Witten's conjecture and Property P

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-coecient 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(3). In particular, Y cannot be a homotopy-sphere.

Recurrence relations and asymptotics for four-manifold invariants

The polynomial invariants

Knot homology groups from instantons

q_d

Instanton Floer homology and the Alexander polynomial

for a large class of smooth 4-manifolds are shown to satisfy universal relations. The relations reflect the possible genera of embedded surfaces in the 4-manifold and lead to a structure theorem for the polynomials. As an application, one can read off a lower bound for the genera of embedded surfaces from the asymptotics of

Gauge theory and Rasmussen's invariant

q_d

The genus-minimizing property of algebraic curves

for large

Filtrations on instanton homology

d

The Geometry of Four-Manifolds

. The relations are proved using moduli spaces of singular instantons.

The Genus of Embedded Surfaces in the Projective Plane

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.

Gauge theory for embedded surfaces, II

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.