Resumen de Some phenomena in tautological rings of manifolds

Oscar Randal Williams

• We prove several basic ring-theoretic results about tautological rings of manifolds W, that is, the rings of generalised Miller–Morita–Mumford classes for fibre bundles with fibre W. Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitely-generated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W. Lastly, we give extensive computations in the tautological rings of CP2 and S2 × S2.