Tautological classes of definite 4–manifolds

Baraglia, David

doi:10.2140/gt.2023.27.641

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Tautological classes of definite $4$–manifolds

David Baraglia

Geometry & Topology 27 (2023) 641–698

DOI: 10.2140/gt.2023.27.641

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 ${ℂ ℙ}^{2}$ and ${ℂ ℙ}^{2} # {ℂ ℙ}^{2}$ . 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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Volume 27, issue 2 (2023)