Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

TL;DR

This paper proves that equivariant cohomology Chern numbers fully determine equivariant unitary bordism classes for torus groups, confirming a conjecture and solving related questions in equivariant topology.

Contribution

It establishes that equivariant cohomology Chern numbers uniquely determine equivariant unitary bordism classes for torus groups, confirming a key conjecture and extending to Hamiltonian G-manifolds.

Findings

Equivariant cohomology Chern numbers determine bordism classes.

Confirmed the conjecture by Guillemin--Ginzburg--Karshon.

Extended methods to unoriented bordism and classical results.

Abstract

This paper shows 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 the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--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 $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck.