A result on the (A)over-cap and elliptic genera on non-spin manifolds with circle actions

We prove the vanishing of the (A) over cap -genus of compact smooth manifolds with finite second homotopy group and endowed with smooth S-1 actions. These manifolds Are not necessarily spin, hence, this vanishing extends that of Atiyah and Hirzebruch on spin manifolds with S-1 actions. The proof is accomplished by proving a rigidity theorem under circle actions of the elliptic genus on these manifolds.