Some phenomena in tautological rings of manifolds

TL;DR

This paper investigates the algebraic structure of tautological rings associated with manifolds, establishing conditions for finite generation, exploring the influence of torus actions, and providing explicit computations for specific manifolds.

Contribution

It offers new criteria for finite generation of tautological rings, links torus symmetries to algebraic properties, and presents detailed calculations for complex projective and product of spheres.

Findings

Tautological rings can be finitely generated under certain cohomological conditions.

Torus actions impose lower bounds on the Krull dimension of tautological rings.

Explicit computations for CP^2 and S^2 x S^2 illustrate the theoretical results.

Abstract

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 CP^2 and S^2 x S^2.