JGS Friday April 1st 11 am

[Rafael Torres]

Date/time/place Friday April 1st/11 am/Room 136

Title: Knots and smooth structures on 4-manifolds

Speaker: Oliviero Malech (SISSA)

Abstract: The objective of the talk is to show how knot theory and 4-manifold topology can interact.
I will describe the Fintushel-Stern knot surgery on a smooth closed oriented 4-manifold X, a construction that involves a (tame) knot K in S^3 and an embedded 2-torus in X. The result of this procedure is a new smooth oriented 4-manifold, X_K.
The principal theorem will be that under some additional assumption  X and X_K are homeomorphic and their Seiberg-Witten invariants differ exactly by the Alexander polynomial of the knot K.
Therefore by this theorem is possible to construct on the same smooth 4-manifold inequivalent smooth structures. The procedure shows that the K3 surface admits infinitely many inequivalent smooth structures.