Tautological classes of definite 4-manifolds

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 ${\mathbb{CP}}^2$ and ${\mathbb{CP}}^2\#{\mathbb{CP}}^2$. 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.

arXiv comment46 pages, to appear in Geometry & Topology

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