Tautological Rings of Fibrations

TL;DR

This paper investigates tautological rings in fibrations with Poincaré fibers, providing methods to compute these rings and explicitly determining them for certain classical spaces.

Contribution

It introduces a framework for computing tautological rings in fibrations and explicitly determines these rings for even spheres, complex projective spaces, and some products of odd spheres.

Findings

Computed the Euler ring using rational homotopy theory tools.

Determined the tautological ring for even spheres and complex projective spaces.

Extended the analysis to some products of odd spheres.

Abstract

We study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincar\' e fibre, i.e. the ring obtained by fibre integrating powers of the fibrewise Euler class. We discuss how to compute the Euler ring with tools from rational homotopy theory and completely determine the tautological ring for even spheres, complex projective spaces and some products of odd spheres.