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Tautological classes of definite $4$–manifolds

David Baraglia

Geometry & Topology 27 (2023) 641–698
Abstract

We prove a diagonalisation theorem for the tautological, or generalised Miller–Morita–Mumford, classes of compact, smooth, simply connected, definite 4–manifolds. Our result can be thought of as a families version of Donaldson’s diagonalisation theorem. We prove our result using a families version of the Bauer–Furuta cohomotopy refinement of Seiberg–Witten theory. We use our main result to deduce various results concerning the tautological classes of such 4–manifolds. In particular, we completely determine the tautological rings of 2 and 2 # 2. We also derive a series of linear relations in the tautological ring which are universal in the sense that they hold for all compact, smooth, simply connected definite 4–manifolds.

Keywords
tautological classes, Miller–Morita–Mumford classes, Seiberg–Witten, Bauer–Furuta, definite 4–manifolds
Mathematical Subject Classification
Primary: 53C07, 57K41, 57R22
References
Publication
Received: 17 August 2020
Revised: 2 September 2021
Accepted: 5 October 2021
Published: 16 May 2023
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Nathalie Wahl
Authors
David Baraglia
School of Mathematical Sciences
The University of Adelaide
Adelaide SA
Australia

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