The signature and the elliptic genus of π2-finite manifolds with circle actions

We prove the vanishing of various characteristic numbers, such as the signature and the A-genus, on manifolds with finite second homotopy group and which admit smooth S1-actions. More precisely, we prove the vanishing of various coefficientes of the elliptic genus on non-spin π2-finite manifolds when the S1-action either satisfies a “parity” condition or has isolated fixed points only.