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Type: Journal article
Title: On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds
Author: Baraglia, D.
Konno, H.
Citation: Journal of Topology, 2022; 15(2):505-586
Publisher: London Mathematical Society
Issue Date: 2022
ISSN: 1753-8416
Statement of
David Baraglia, Hokuto Konno
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 spinc 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.
Rights: © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
DOI: 10.1112/topo.12229
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Appears in Collections:Mechanical Engineering publications

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