On the Equivalence of Integral T^k-Cohomology Chern Numbers and T^k-K-Theoretic Chern Numbers

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant $K$-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant $K$-theoretic Chern numbers.