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Rigidity of the mod 2 families Seiberg–Witten invariants and topology of families of spin 4-manifolds
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 4 / April 2021
- Published online by Cambridge University Press:
- 15 April 2021, pp. 770-808
- Print publication:
- April 2021
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- Article
- Export citation
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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 
$\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 
$4$-manifolds.
