ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

6. Topology

1991 MS Classification: 57R55 Fintushel, Ronald, Michigan State University, USA; Stern, Ronald J., University of California at Irvine, USA Knot Surgery on 4-Manifolds

In this talk we shall discuss geometric operations on a given smooth 4manifold which preserve the underlying topological structure and alter its smooth structure. In particular, we shall consider the following construction: Let X be a simply connected smooth 4-manifold which contains a smoothly embedded torus T of self-intersection 0. Given a knot K in S^{3}, we replace a tubular neighborhood of K with S^{1} × (S^{3} \ K) to obtain the knot surgery manifold X_{K}. More formally, this procedure is accomplished by performing 0-framed surgery on K to obtain the 3-manifold M_{K}. The meridian m of K can be viewed as a circle in M_{K}; so in S^{1} × M_{K} we have the smooth torus T_{m} = S^{1} × m of self-intersection 0. Since a neighborhood of m has a canonical framing in M_{K}, a neighborhood of the torus T_{m} in S^{1} × M_{K} has a canonical identification with T_{m} × D^{2}. The knot surgery manifold X_{K} is given by the fiber sum

X_{K} = X#_{T =T_{m}}S^{1} × M_{K} = (X \ T × D^{2}) ∪ (S^{1} × M_{K} \ T_{m} × D^{2})

where the two pieces are glued together so as to preserve the homology class [pt × ∂D^{2}]. This latter condition does not, in general, completely determine the diffeomorphism type of X_{K}, and X_{K} is taken to be any manifold constructed in this fashion. We shall discuss consequences of this knot surgery construction. For example, under relatively mild hypotheses on (X, T), X_{K} will be homeomorphic to X, and its Seiberg-Witten invariant (viewed as a Laurent polynomial) will be SW_{X_{K}} = SW_{X} ·Δ_{K}(t), where Δ_{K}(t) is the symmetrized Alexander polynomial of K in the variable t = exp(2[T_{m}]). As a consequence, given (X, T) satisfying the hypotheses, this produces infinitely many nondiffeomorphic 4-manifolds, all homeomorphic to X. In particular, this works for (K3, elliptic fiber). If the knot K is fibered, and X is symplectic with T a symplectically embedded torus, then the knot manifold X_{K} is also symplectic. However, if the Alexander polynomial of K is not monic, the knot manifold X_{K} supports no symplectic structure.

We shall also use a similar technique to construct exotic embedded surfaces in 4-manifolds, and discuss what is known about geography of smooth 4manifolds. Reference: R. Fintushel and R. Stern, Knots, links, and 4-manifolds, to appear in Invent. Math.