The Homology of Complex Equivariant Bordism

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 $mU_A$.