Algebraic Cobordism of Classifying Spaces

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).