One Fixed Point Actions and Homology 3-Spheres

Abstract

One of the best known three-dimensional manifolds is the Poincare homology three-sphere 1(2, 3,5); this closed 3-manifold has the same homology groups as the standard sphere S3 but is not simply connected (see [Pnc, p. 106] or [Br2, Section 1.8]). Numerous investigations during the past century have shown that this manifold has many remarkable properties; several equivalent descriptions of 1(2,3,5) are summarized in [KiS]. From the viewpoint of transformation groups, one noteworthy property is that 1(2, 3, 5) is the only nonsimply connected homology sphere admitting a transitive action of a compact Lie group [Brl]. The manifold 1(2, 3,5) also figures importantly in regularity questions for group actions on spherelike manifolds. Linear (or orthogonal) actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions, and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions. One distinguishing feature of linear actions is that their fixed point sets (if nonempty) are also spheres. The well-known results of P. A. Smith (cf. [Br2]) show that continuous p-group actions on homology spheres all resemble linear actions in this respect; i.e., the fixed point sets are always Zp-homology (cohomology) spheres. On the other hand, outside the family of p-groups, there are many exotic group actions. Perhaps the most elementary example of this type involves 1(2, 3,5). If A is the subgroup of the rotation group SO3 given by the symmetries of a regular icosahedron (so that A is isomorphic to the alternating group A5), then the quotient space S03/A is the Poincare homology 3-sphere and the induced action of A5 ? A has exactly one fixed point. This example suggests that general actions of finite groups on homology spheres can be quite different from linear actions, and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action.