Actions of finite abelian groups of odd prime power order

Abstract

We now deal with problems of p odd similar to those of Chapter V for p = 2. We lead off in section 43 with differentiable, orientation preserving actions of (Z _p )^k, p an odd prime, on closed oriented manifolds V ⁿ. The primary aim is to give existence theorems for stationary points of such actions. Our interest in such problems has been aroused by the work of Borel [6, 9], although we attack the problem from a different point of view. An example of a corollary of our results is that if V ⁿ has one of its Pontryagin numbers not divisible by p then the action has a stationary point. We go on to note that if a toral group acts on V ⁿ without stationary points then [V ⁿ] represents a torsion element of Ω _n .