Complex \(N\)-spin bordism of semifree circle actions and complex elliptic genera (Q303937)

An \(S^1\)-action on a space \(M\) is called \textit{semifree} if it is free on the complement of the fixed points. If \(M\) is a spin-manifold and the action is smooth, we say that the action is of \textit{odd type} if the action on the \(\mathrm{SO}(n)\)-bundle associated to the tangent bundle \(TM\) does not lift to its \(\mathrm{Spin}(n)\)-cover. In [Topology 26, 143--151 (1987; Zbl 0626.57014)] \textit{S. Ochanine} showed that the ideal in the oriented bordism ring \(\Omega^{\mathrm{SO}}_*\otimes \mathbb{Q}\) consisting of rational multiples of bordism classes of spin manifolds with a semifree \(S^1\)-action of odd type agrees with the kernel of the elliptic genus; this allowed to describe the generators of this ideal, answering a question of Landweber and Stong. The rigidity theorem for the elliptic genus implies that we can replace the condition to be semifree with being effective without changing the ideal.NEWLINENEWLINEAssume that \(M\) is stably almost complex, i.e. that we choose a complex structure on \(TM\) after adding a trivial bundle. The choice of an additional spin structure is then equivalent to choosing a root of the complex determinant line bundle of \(TM\). We say that \(M\) is \textit{\(N\)-spin} if we choose an \(N\)-th root of this determinant line bundle. To an \(S^1\)-action on an \(N\)-spin manifold we can associate a type in \(\mathbb{Z}/N\).NEWLINENEWLINEWhile the original elliptic genus is based on modular forms for the congruence subgroup \(\Gamma_1(2)\), Hirzebruch introduced elliptic genera \(\phi_n\) based on modular forms for the congruence subgroups \(\Gamma_1(n)\) for all \(n\geq 2\). He showed that \(\phi_n([M]) =0\) if \(M\) is an \(n\)-spin manifold with an effective \(S^1\)-action of nonzero type. Let \(I^{N,t}\) be the ideal in \(MU_*\otimes \mathbb{Q}\) of all rational multiples of \(N\)-spin manifolds with an effective \(S^1\)-action of type \(t\). In [``Complex elliptic genera and \(S^1\)-equivariant cobordism theory'', Preprint, \url{arXiv:math/0405232}], \textit{G. Höhn} conjectured for nonzero \(t\) that \(I^{N,t}\) consists of exactly those elements where \(\phi_n\) vanishes for every \(n\) dividing \(N\) but not \(t\) and showed this for \(t = [1]\).NEWLINENEWLINEThe main contribution of the paper under review is to prove this conjecture in most cases if \(N\leq 9\). The proofs are partially based on computer calculations. The paper contains further results about bordism of manifolds with semifree \(S^1\)-action.