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 spin${}^c$ 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.

76 pages, accepted version. To appear in J. Topology

References in corpus (2)

• Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds
• A note on the Nielsen realization problem for K3 surfaces

Cited by in corpus (8)

• Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants
• Isotopy of the Dehn twist on K3#K3 after a single stabilization
• Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds
• A note on the Nielsen realization problem for K3 surfaces
• Constraints on families of smooth 4-manifolds from ${\mathrm{Pin}}^{-}(2)$-monopole
• Tautological classes of definite 4-manifolds
• Non-trivial smooth families of $K3$ surfaces
• Exotic embedded surfaces and involutions from Real Seiberg-Witten theory