Complex $N$-Spin bordism of semifree circle actions and complex elliptic genera

摘要

We give a complete bordism analysis of rational bordism groups of semifree circle actions on complex $N$-Spin manifolds. Moreover, we introduce the notion of a complex $N$-Spin$^{c,t}$ manifold and give a characterization of cobordism groups of such manifolds which we use to compute the rational bordism groups of free circle actions of type $t$ on complex $N$-Spin manifolds. Furthermore, we exploit this bordism analysis to furnish a mechanism with which we investigate a description, in terms of kernels of complex elliptic genera, of the ideal $I_*^{N,t}$, generated by bordism classes of connected complex $N$-Spin manifolds admitting an effective circle action of type $t$, in the rational complex $N$-Spin cobordism ring $Omega_*^{U,N}otimesmathbb{Q}$.