Inequivalent, bordant group actions on a surface

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with (g) orientation-preserving for all g e G. In addition we will consider only effective ( is injective) free actions. Free means that (g) is fixed-point-free for all geG, g 4= 1. This paper addresses the classification of such actions. Given an action of G on F, as above, new actions can be constructed by conjugating with homeomorphisms of F. Hence, two actions, (p1 and 02, are called equivalent if there is a homeomorphism h of F such that $%(g) = hox{g) oh, for all geG. One invariant of equivalence classes of actions is their free C-bordism class. Essentially, G actions on surfaces Fj and F2 are freely Cr-bordant if there is a compact, orientable 3-manifold, M, with boundary Fx u — F2, such that there is a free orientationpreserving action of G on M restricting to the two given actions on the boundary components of M. (See [2] for details.) The set of Cr-bordism classes forms a group, denoted d%(G), which is isomorphic to H2(G). It follows from the definitions that equivalent actions are freely bordant. The point of this paper is that the converse does not hold. We will construct an example of two inequivalent actions of the symmetric group, Sa, on a surface such that both actions are null bordant. Although it seemed unlikely that bordism could classify group actions, partial results in this direction have been obtained. In particular, freely bordant actions are equivalent in the case that G is cyclic [6], abelian [3], or metacyclic [4]. Related results have also been proved for the unoriented case [8,9]. Furthermore, freely bordant actions are equivalent in a fairly strong stable sense [5]. Thanks are due here to Allan Edmonds, for pointing out this problem to me and for many valuable conversations, as well as to Charlie Frohman and Richard Skora.