Semifree actions on homotopy spheres

Abstract

In this paper, we study the semifree Z actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree Z4s actions on even dimensional homotopy sphere pairs. 0. Introduction. Let G be a compact Lie group. A differentiable group action of G on a differentiable manifold Mn is a homomorphism i/i: GDiff (M), where Diff (M) is the group of diffeomorphisms of M. Let Fk be the submanifold of fixed points. A group action is semifree if the only isotropy subgroups are the trivial subgroup and the group itself. Under these restrictions, the action is linear in some neighborhood of Fk in M' in the sense that there is an equivariant vector bundle v normal to Fk in M' such that the action restricts on each fiber of v to a linear automorphism. In this paper, we are only interested in the differentiable semifree actions on homotopy sphere pairs, namely, the actions on homotopy spheres such that the fixed point sets are homotopy spheres. The action G = S1 defines a complex structure on v (the action on v is just that induced by the complex multiplication). In [2], Browder proved that for a semifree S1 action on a homotopy sphere pair, the normal bundle of the fixed point set is stably trivial as a complex vector bundle. The first question which interests us is the following: Problem 1. Is it true that the equivariant normal bundle to the fixed point set of a semifree Zm action on a homotopy sphere pair is equivariantly stably trivial? If G = Sl or Zm, it is known (see [2], [6]) that there are infinitely many semifree G actions on odd dimensional homotopy sphere pairs. As to semifree S1 actions on even dimensional homotopy sphere pairs, there are only finitely many [3]. Received by the editors February 22, 1974 and, in revised form, July 9, 1974. AMS (MOS) subject classifications (1970). Primary 57E25, 57E30.