Constraints on families of smooth 4-manifolds from Bauer-Furuta 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 $8$.

arXiv comment25 pages, revised version with some corrections and an improvement (removed a hypothesis in the case of positive first Betti number). To appear in Algebr. Geom. Topol

References in corpus (5)

• Seiberg-Witten Theory and Z/2^p actions on spin 4-manifolds
• On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds
• Equivalences of Smooth and Continuous Principal Bundles with Infinite-Dimensional Structure Group
• Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory
• Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

Cited by in corpus (7)

• Isotopy of the Dehn twist on K3#K3 after a single stabilization
• Equivariant Seiberg-Witten-Floer cohomology
• 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
• Dehn twists and the Nielsen realization problem for spin 4-manifolds
• Exotic embedded surfaces and involutions from Real Seiberg-Witten theory