Free actions of nite groups on rational homology 3 - spheres

The purpose of this note is to prove the following: Theorem 1.1 Let G be a nite group. Then there is a rational homology S 3 on which G acts freely. That any nite group acts freely on some closed 3-manifold is easy to arrange: There are many examples of closed 3-manifolds whose fundamental groups surject a free group of rank two (for example, by taking a connected sum of S 1 S 2 's) and by passing to a covering space, one can obtain a manifold whose group surjects a free group of any given rank. This gives a surjection onto any nite group and hence a free action on the associated covering space. We also note that results of Milnor 2] easily imply that one cannot replace rational coeecients by integral coeecients and hope for a similar result. The strategy for proving Theorem 1.1 is this: We begin with a free action of G on some 3-manifold M. This makes H 1 (M) into a representation module for the group G. (Here, as throughout, homology groups will be with rational coeecients.) Our rst task is to gain some control over the representations which occur. To this end we recall that every nite group acts on its rational group algebra QG] by left multiplication to give the so-called left regular representation. We denote this representation by L G. Then the control we seek is accomplished in Lemma 2.3, where, denoting the trivial representation by < 1 > (that is to say, the one dimensional vector space with the trivial G-action) we show that one can nd a possibly diierent 3-manifold and a free G-action, so that the G-module H 1 (M) < 1 > becomes a large number of copies of L G. We then show that one can systematically remove summands of this controlled type by Dehn surgery, a process which eventually yields a rational homology sphere with free G action. We conclude with a sketch that this rational homology sphere can be chosen to be hyperbolic. 2 The construction. Suppose that M is a 3-manifold with a free G-action. Suppose that 1 ; : : : ; k is a set of disjoint smooth simple closed curves in M which are freely permuted by G: Equivariantly deleting open regular neighbourhoods of these curves, we form the manifold X = M n G N(1 t : : : …