Tautological classes of definite 4–manifolds

Baraglia, David¹

¹ School of Mathematical Sciences, The University of Adelaide, Adelaide SA, Australia

Geometry & topology, Tome 27 (2023) no. 2, pp. 641-698.

Résumé

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 ${ℂ\mathbb{P}}^2$ and ${ℂ\mathbb{P}}^2\#{ℂ\mathbb{P}}^2$. 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.

Détail

DOI : 10.2140/gt.2023.27.641

Keywords: tautological classes, Miller–Morita–Mumford classes, Seiberg–Witten, Bauer–Furuta, definite 4–manifolds

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