Almost free actions on manifolds

Abstract

Let X be a compact, connected, oriented topological G -manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U (1) = S 1 and dim X =4 or G = S p (1) = S 3 and dim X = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, S p (1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and (2) the (even) number of fixed points. Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U (1) is also proved.