The Homology of Complex Equivariant Bordism

TL;DR

This paper computes the $E$-homology of equivariant complex bordism spectra for abelian compact Lie groups, correcting previous errors and providing new calculations for both homotopical and geometric versions.

Contribution

It provides the first corrected and comprehensive computation of $E$-homology for $A$-equivariant complex bordism spectra, including both homotopical and geometric cases.

Findings

Corrected previous errors in the computation of $E$-homology.

Calculated $E$-homology for the geometric $A$-equivariant complex bordism spectrum.

Established new formulas for the $E$-homology of $MU_A$ and $mU_A$.

Abstract

Let $A$ be an abelian compact Lie group and let $E$ be an oriented $A$-spectrum. We compute the $E$-homology of tom Dieck's homotopical $A$-equivariant complex bordism spectrum $MU_A$ in two ways, correcting an error in Cole-Greenlees-Kriz (2002). Additionally, we calculate the $E$-homology of the geometric $A$-equivariant complex bordism spectrum $mU_A$.