Circle actions on spin manifolds and characteristic numbers

Abstract

II+ 1970 Atiyah and Hirzebruch [3] proved that the existence of a non-trivial smooth action of the circle group S’ on a closed connected Spin manifold M implies that the A-genus of M vanishes. It is our purpose to explore the vanishing of further Pontrjagin numbers of Spin manifolds admitting smooth S’ actions of odd type; assuming that MS’ # 4, and viewing Z, = ( k 1) c S’, these are the actions for which all components of the fixed set MzL have codimensions congruent to 2 (mod 4) (see [2,3]). We have one general result for such actions; the rest of our results require that the action also be semifree, i.e. that it be free on the complement of the fixed set MS’. The following result was proved during conversations with S. Weinberger.