A bordism theory related to matrix Grassmannians

TL;DR

This paper develops a bordism theory involving manifolds with trivial normal bundles and virtual SU-bundles, calculating its ring structure and generators.

Contribution

It introduces a new bordism theory related to pairs of manifolds and virtual SU-bundles, providing explicit calculations of its ring structure.

Findings

Ring modulo torsion computed

Explicit generators described

Bordism classes characterized

Abstract

In the present paper we study a bordism theory related to pairs $(M,\, \xi),$ where $M$ is a closed smooth oriented manifold with a stably trivial normal bundle and $\xi$ is a virtual $\SU$-bundle of virtual dimension 1 over $M$. The main result is the calculation of the corresponding ring modulo torsion and the explicit description of its generators.