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 CP2 and CP2 # CP2 . 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.