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

## TL;DR

This paper establishes a cohomological relationship between families Seiberg-Witten and Bauer-Furuta invariants, revealing new obstructions and properties of families of 4-manifolds, with applications to detecting non-smoothable families and proving wall crossing formulas.

## Contribution

It introduces a cohomological formula linking the families Seiberg-Witten and Bauer-Furuta invariants, and explores their properties, including Steenrod squares and K-theoretic aspects.

## Key findings

- Families Seiberg-Witten invariants can be derived from Bauer-Furuta invariants.
- New mod 2 relations between invariants and Chern classes are established.
- A non-smoothable family of K3 surfaces is detected.

## 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.