Cobordism theory of semifree circle actions on complex n-spin manifolds.

Abstract

In this work, we study the complex N-Spin bordism groups of semifree circle actions and elliptic genera of level N. The notion of complex N-Spin manifolds (or simply N-manifolds) was introduced by Hoehn in [Hoh91]. Let the bordism ring of such manifolds be denoted by U;N and the ideal in U;N Q generated by bordism classes of connected complex N-Spin manifolds admitting an e ffective circle action of type t be denoted by IN;t. Also, let the elliptic genus of level n be denoted by 'n. It is conjectured in [Hoh91] that IN;t = \ njN n - tker('n): Our work gives a complete bordism analysis of rational bordism groups of semifree circle actions on complex N-Spin manifolds via traditional geometric techniques. We use this analysis to give a determination of the ideal IN;t for several N and t, and thereby verify the above conjectural equation for those values of N and t. More precisely, we verify that the conjecture holds true for all values of t with N 9, except for case (N; t) = (6; 3) which remains undecided. Moreover, the machinery developed in this work furnishes a mechanism with which to explore the ideal INt for any given values of N and t.