Tautological rings and stabilisation

TL;DR

This paper introduces a ring homomorphism that compares the tautological rings of a smooth manifold and its stabilization by a product of spheres, advancing understanding of their algebraic structures.

Contribution

It constructs a novel ring homomorphism linking the tautological rings before and after stabilization, providing new insights into their algebraic relationships.

Findings

Established a homomorphism between tautological rings of stabilized manifolds

Demonstrated structural similarities in tautological rings under stabilization

Provided tools for further exploration of tautological ring properties

Abstract

We construct a ring homomorphism comparing the tautological ring, fixing a point, of a closed smooth manifold with that of its stabilisation by $S^{2a} \times S^{2b}$.