Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

Abstract

We show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$–manifolds, which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon (Moment maps, cobordisms, and Hamiltonian group actions, Remark H.5 in Appendix H.3), where $G$ is a torus. As a further application, we also obtain a satisfactory solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian $G$–manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber, Panov and Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of ${\left({\mathbb{Z}}_2\right)}^k$–equivariant unoriented bordism and can still derive the classical result of tom Dieck.