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Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

Zhi Lü and Wei Wang

Algebraic & Geometric Topology 18 (2018) 4143–4160

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 (2)k–equivariant unoriented bordism and can still derive the classical result of tom Dieck.

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