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.
Categories: math.DG math.GT