Tautological rings for high dimensional manifolds

TL;DR

This paper investigates the structure of tautological rings associated with high-dimensional manifolds, specifically describing these rings modulo nilpotent elements for connected sums of $S^n imes S^n$ when $n$ is odd.

Contribution

It provides a complete description of tautological rings modulo nilpotent elements for certain high-dimensional manifolds, advancing understanding of their algebraic structure.

Findings

Tautological rings are characterized modulo nilpotent elements.

Explicit description for connected sums of $S^n imes S^n$ when $n$ is odd.

Enhanced understanding of characteristic classes in high-dimensional topology.

Abstract

We study tautological rings for high dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^*(M)$ of those of characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller--Morita--Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^n \times S^n$ for $n$ odd.