Algebraic Cobordism of Classifying Spaces

TL;DR

This paper develops algebraic cobordism theories for classifying spaces of algebraic groups, computes these for several classical and finite groups, and verifies their isomorphism with complex cobordism.

Contribution

It introduces algebraic cobordism of classifying spaces and computes it for various algebraic groups, establishing key properties and isomorphisms.

Findings

alculated algebraic cobordism  classifying spaces for classical Lie groups.

alculated algebraic cobordism  finite abelian groups and quaternion group.

ound obordism  classifying spaces isomorphic to MU^*(BG).

Abstract

We define algebraic cobordism of classifying spaces, \Omega^*(BG) and G-equivariant algebraic cobordism \Omega^*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(\Omega^*(-)), which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We calculate \Omega^*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n+1). We also calculate \Omega^*(BG) when G is a finite abelian group. A finite non-abelian group for which we calculate \Omega^*(BG) is the quaternion group of order 8. In all the above cases, we check that \Omega^*(BG) is isomorphic to MU^*(BG).