Rigidity of elliptic genera for nonspin manifolds

We discuss the rigidity of elliptic genera for nonspin 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 2 .M / that guarantees the rigidity in the case that the universal covering of M is nonspin.53C27, 57R15, 57R91, 57S15, 58J26 Michael WiemelerWe call S 1 OEM rigid if a nk D 0 whenever k 0. It has been conjectured by Witten [17] that equivariant elliptic genera of closed, spin manifolds with S 1 -action are rigid.Here an oriented manifold is called spin if its second Stiefel-Whitney class vanishes.Witten's conjecture was proved by Bott and Taubes [4].Alternative proofs have been given by Dessai [6], Liu [14], Taubes [15] and Wiemeler [16].Herrera and Herrera [9] proposed a proof of the rigidity of elliptic genera of 2 -finite, oriented manifolds.However, a mistake in the proof was found by Amann and Dessai [1].They suggested that assuming 2 -and 4 -finiteness would be enough to come around this problem and to prove the rigidity (see the correction to [9]).The aim of this paper is to clarify Lemma 1 of [9].We find a counterexample to the original statement and obtain the same conclusion under additional hypothesis.We note that Amann and Dessai clarified the proof of Lemma 2 of [9].Lemma 1 of [9] claims that if M is an even-dimensional orientable manifold acted on by S 1 , then the Hfixed point components in M which contain S 1 -fixed points are orientable for every subgroup H S 1 .We prove this claim under the extra assumption that the universal covering of M is a spin manifold.This then implies the rigidity of the elliptic genera of closed, oriented S 1 -manifolds M with spin universal cover.We note that if the oriented, closed, connected manifold M has a finite cover z M which is spin then the rigidity of S 1 OEM can be deduced directly from the Bott-Taubes theorem.Indeed, by the Lefschetz fixed point formula we can assume that there are S 1 -fixed points in M.This implies that the action of S 1 on M lifts to an action on z M.Moreover, since z M !M is an equivariant covering map with k < 1 sheets the equivariant elliptic genera of z M and M are related as follows:Therefore the rigidity for M follows from the rigidity for z M.However, we also note that there are manifolds with spin universal cover which do not admit any finite spin cover (see Ebert's explanation [7]).Therefore we think it is worth stating and proving the following:Theorem 1.1 Let M be a closed , oriented , even-dimensional S 1 -manifold whose universal covering is spin.Then the equivariant elliptic genus of M is rigid.Using an argument which goes back to Hirzebruch and Slodowy [11] one sees that this theorem implies the following generalization of a result of Atiyah and Hirzebruch [2].Corollary 1.2 Let M be a closed , connected , oriented manifold whose universal covering is spin such that S 1 acts nontrivially on M. Then the O A-genus of M vanishes.In [1] (see also the addendum) examples of oriented, closed, simply connected, effective S 1 -manifolds M with 2 .M / D Z 2 and O A.M / 0 were given.In view of the original claim in [9] and Corollary 1.2 one might ask if there is any condition on 2 of a closed, connected, oriented, effective S 1 -manifold M