Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

Abstract

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\mathrm{Diff}(M)\hookrightarrow \mathrm{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 4-manifolds.

40 pages, to appear in Compositio Mathematica

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• On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds
• Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants
• A note on the Nielsen realization problem for K3 surfaces