Local surgery obstructions and space forms

Abstract

The space form problem, concerning the existence of free actions of finite groups on spheres, has now been substantially solved except in dimension three I-3, 4, 8, 11, 13, 14]. This paper gives an effective method for the uniqueness question: which homotopy types are realized by such actions. Our results are fairly complete for 2-hyperelementary periodic groups and give new necessary conditions for actions even on S 3. In particular Corollary C below eliminates certain homotopically non-linear actions which presented an obstacle to knowing precise dimensional bounds (see [4; Conjecture D]). The methods also apply to the euclidean space form problem of semi-free actions on I( "+k fixing I( k [2] where the linear model is a free representation direct sum with a trivial representation. A free action on S'- 1 yields a semi-free topological action on l(" by "coning". The method involves the calculation of a certain local surgery obstructions (w167 2, 3). The basic cases are the 2-hyperelementary type I groups: extensions of the form l ~ Z/m~ ~--, Z/2k--> l where m is odd. We write o- for a Sylow 2-subgroup of ~z (and identify it with Z/2k). Let t be the twisting defining the extension, and let t(cr) = Im(t: Z/2 k --* Aut(Z/m)) -~ 7/./2 z.