Rigidity of elliptic genera for non-spin manifolds

We discuss the rigidity of elliptic genera for non-spin manifolds $M$ with $S^1$-action. We show that if the universal covering of $M$ is spin, then the universal elliptic genus of $M$ is rigid. Moreover, we show that there is no condition which only depends on ${\pi }_2(M)$ that guarantees the rigidity in the case that the universal covering of $M$ is non-spin.