Algebraic cobordism of varieties with G-bundles

TL;DR

This paper extends algebraic cobordism theory to varieties with principal G-bundles for reductive groups, showing it aligns with standard cobordism after inverting the torsion index and relating it to classifying spaces.

Contribution

It demonstrates that the cobordism theory for G-bundles extends standard algebraic cobordism rationally and after inverting the torsion index, and relates the theory for G to that for a maximal torus T.

Findings

The theory for bundles on varieties extends standard algebraic cobordism after inverting the torsion index.

The theory for a point is dual to Omega^*(BG).

The theory for G relates to that for T (a maximal torus).

Abstract

Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory for G compares to that for a maximal torus T.