Tautological classes of definite 4-manifolds

TL;DR

This paper establishes a diagonalisation theorem for tautological classes of definite 4-manifolds, using advanced Seiberg-Witten theory, and determines their tautological rings for specific cases like P^2.

Contribution

It introduces a families version of Donaldson's diagonalisation theorem and applies it to explicitly compute tautological rings of certain 4-manifolds.

Findings

Complete determination of tautological rings for P^2 and P^2 \u2227 P^2.

Derivation of universal linear relations in tautological rings.

Application of families Seiberg-Witten theory to 4-manifold topology.

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.