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.

Non-trivial smooth families of $K3$ surfaces

“…If w(η) and w(s) are disjoint, then (w(η) − w(s))/||w(η) − w(s)|| H defines a section of S(H). The wall crossing formula for the families Seiberg-Witten invariants [12], [3] adapted to the present setting states that…”

“…Let Gr 3 (R 3,19 ) denote the Grassmannian of positive definite 3-planes in R 3,19 . There is a period map P : T Ein → Gr 3 (R 3,19 ). Defined as follows.…”

“…Defined as follows. Fix an isometry H 2 (X; R) ∼ = R 3,19 . Then P sends an Einstein metric g to the 3-plane…”

“…We take E δ → S 2 to be the pullback of the family E Ein → T Ein by a map g δ : S 2 → T Ein . Let P : T Ein → Gr 3 (R 3,19 ) be the period map. Then f δ = P • g δ : S 2 → Gr 3 (R 3,19 ) is defined as…”

“…Let P : T Ein → Gr 3 (R 3,19 ) be the period map. Then f δ = P • g δ : S 2 → Gr 3 (R 3,19 ) is defined as…”