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Equivariant unitary bordism and equivariant cohomology Chern numbers

Research paper by Zhi Lü, Wei Wang

Indexed on: 09 Aug '14Published on: 09 Aug '14Published in: Mathematics - Algebraic Topology



Abstract

By using the universal toric genus and the Kronecker pairing of bordism and cobordism,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 ~\cite[Remark H.5, \S3, Appendix H]{Gullemin}, where $G$ is a torus. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. As a further application, we also obtain a satisfactory solution of~\cite[Question (A), \S1.1, Appendix H]{Gullemin} on unitary Hamiltonian $G$-manifolds. In particular, our approach can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism, and without the use of Boardman map, it can still work out the classical result of tom Dieck, which states that the $({\Bbb Z}_2)^k$-equivariant unoriented bordism class of a smooth closed $({\Bbb Z}_2)^k$-manifold is determined by its $({\Bbb Z}_2)^k$-equivariant Stiefel--Whitney numbers. In addition, this paper also shows the equivalence of integral equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers for determining the equivariant unitary bordism classes of closed unitary $G$-manifolds by using the developed equivariant Riemann--Roch relation of Atiyah--Hirzebruch type, which implies that, in a different way, we may induce another classical result of tom Dieck, saying that equivariant K-theoretic Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds.