Algebraic & Geometric Topology

Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

Zhi Lü and Wei Wang

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4143-4160.

Received: 10 December 2017
Revised: 23 April 2018
Accepted: 10 June 2018
First available in Project Euclid: 18 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 57R20: Characteristic classes and numbers 57R85: Equivariant cobordism 57R91: Equivariant algebraic topology of manifolds

equivariant unitary bordism Hamiltonian bordism equivariant cohomology Chern number


Lü, Zhi; Wang, Wei. Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups. Algebr. Geom. Topol. 18 (2018), no. 7, 4143--4160. doi:10.2140/agt.2018.18.4143.

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