Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ 1 and φ 2 of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [ M , φ 1 ] = [ M , φ 2 ] in the oriented bordism group Ω 2 ( G ) of the group G. In this paper, we compute the bordism class [ M , φ ] for any such action of G on M and we determine for a given M, the bordism classes in Ω 2 ( G ) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form ( Z / p α Z ) n ) .