On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds

Abstract

We show how the families Seiberg–Witten invariants of a family of smooth 4-manifolds can be recovered from the families Bauer–Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg–Witten invariants. We give a formula for the Steenrod squares of the families Seiberg–Witten invariants leading to a series of mod 2 relations between these invariants and the Chern classes of the spinc index bundle of the family. As a result, we discover a new aspect of the ordinary Seiberg–Witten invariants of a 4-manifold X: they obstruct the existence of certain families of 4-manifolds with fibres diffeomorphic to X. As a concrete geometric application, we shall detect a non-smoothable family of K3 surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce K-theoretic Seiberg–Witten invariants and give a formula expressing the Chern character of the K-theoretic Seiberg–Witten invariants in terms of the cohomological Seiberg–Witten invariants. This leads to new divisibility properties of the families Seiberg–Witten invariants.