#### Volume 18, issue 7 (2018)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

### Zhi Lü and Wei Wang

Algebraic & Geometric Topology 18 (2018) 4143–4160
##### 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 ${\left({ℤ}_{2}\right)}^{k}$–equivariant unoriented bordism and can still derive the classical result of tom Dieck.

##### Keywords
equivariant unitary bordism, Hamiltonian bordism, equivariant cohomology Chern number
##### Mathematical Subject Classification 2010
Primary: 55N22, 57R20, 57R85, 57R91
##### Publication
Received: 10 December 2017
Revised: 23 April 2018
Accepted: 10 June 2018
Published: 11 December 2018
##### Authors
 Zhi Lü School of Mathematical Sciences Fudan University Shanghai China Wei Wang College of Information Technology Shanghai Ocean University Shanghai China