Characteristic numbers and group actions

Abstract

Let G denote the finite cyclic group of order n. The problem of determining necessary and sufficient conditions for F to be the fixed point set of a smooth cyclic group action on some sphere has been solved when n is a prime power. P. A. Smith proved that F must be a Zn-homology sphere. If n is odd it is also known that F is unitary. L. Jones has shown that these conditions are also sufficient to realize F as the fixed point set of smooth cyclic group action on some sphere when n is a prime power Pl. In the general case it is known that F is a union of smooth manifolds, unitary if n is odd. And if Zn acts on some even dimensional sphere, then the Euler characteristic number of F is 2 if action is orientation-preserving and 0 if action is orientation-reversing. We may ask about possible restrictions on Pontryagin numbers of components of the fixed set F. If n is a prime power, the Pontryagin numbers of the fixed point set all vanish by the P. A. Smith theorem and the Hirzebruch signature theorem. It is natural to ask whether such restrictions hold for other types of smooth cyclic groups acting on spheres. In case where n is not a prime power, by work of R. Oliver [01] and its extention of A. Assadi [Asd] and K. Pawalowski [Pa 1-4], we can construct exotic actions on spheres such that Pontryagin classes of fixed point sets do not vanish. But in these examples the Pontryagin numbers all vanish because these actions bound group actions on disks. R. Schultz has shown in [S2] that if G is a cyclic group whose order is not a prime power, then there are smooth actions of G on spheres such that the fixed point sets have nonzero Pontryagin numbers provided the dimension of the fixed point set is greater than 16. Testing to see if the lower bound in dimensions is necessary. There are two possibilities. First, more sophisticated computations might make it possible to remove the restriction on dimensions. Second, there might be some unusual things happening in low dimensions (compare Ewing's result for Zp actions on spheres). Main Theorem is evidence for the first one.