# Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

## TL;DR

This paper develops new topological constraints on families of smooth 4-manifolds using Bauer-Furuta invariants, extending classical theorems and providing examples of non-smoothable group actions.

## Contribution

It introduces a families generalisation of Donaldson's diagonalisation theorem and Furuta's 10/8 theorem, with applications to group actions and diffeomorphism groups.

## Key findings

- Constructs non-smoothable $bZ_p$-actions for any odd prime p.
- Shows the inclusion of diffeomorphisms into homeomorphisms is not a weak homotopy equivalence.
- Provides topological constraints on families of 4-manifolds from Seiberg-Witten invariants.

## Abstract

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for any compact, smooth, simply-connected indefinite $4$-manifold with signature of absolute value greater than $8$.