Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

Lü, Zhi; Wang, Wei

doi:10.2140/agt.2018.18.4143

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

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

DOI: 10.2140/agt.2018.18.4143

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 ${(ℤ_{2})}^{k}$ –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

References

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

Volume 18, issue 7 (2018)