Equivariant unitary bordism and equivariant cohomology Chern numbers
报告人： 吕志教授 （复旦大学数学学院）
报告时间：2015.4.29下午4：00-5:00
报告地点：数学楼二楼学术报告厅
Abstract: In this talk, we consider the essential connection between equivariant unitary bordism and equivariant characteristic numbers. We use the universal toric genus and the Kronecker pairing of bordism and cobordism to show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary G-manifolds, giving an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon, 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 that mixed equivariant characteristic numbers separate equivariant Hamiltonian bordism, which gives a satisfactory solution of the Guillemin--Ginzburg—Karshon’s question on unitary Hamiltonian G-manifolds. In particular, our approach can also be applied to the study of (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 (Z_2)^k-equivariant unoriented bordism class of a smooth closed (Z_2)^k-manifold is determined by its (Z_2)^k-equivariant Stiefel--Whitney numbers.
In addition, we may also prove 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. This is a joint work with Wei Wang.
In addition, we may also prove 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. This is a joint work with Wei Wang.