Summary: Free actions of finite groups on rational homology 3spheres.
D. Cooper and D.D. Long \Lambda
April 26, 1996
The purpose of this note is to prove the following:
Theorem 1.1 Let G be a finite group. Then there is a rational homology S 3 on which G acts freely.
That any finite group acts freely on some closed 3manifold is easy to arrange: There are many
examples of closed 3manifolds whose fundamental groups surject a free group of rank two (for
example, by taking a connected sum of S 1 \Theta 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 finite group and hence a free action on the associated covering space. We also note that results
of Milnor  easily imply that one cannot replace rational coefficients by integral coefficients 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 coefficients.) Our first task is to gain some control over the
representations which occur. To this end we recall that every finite group acts on its rational group
algebra Q[G] by left multiplication to give the socalled left regular representation. We denote this
representation by LG . 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