Free Cyclic Actions on Surfaces and the Borsuk–Ulam Theorem

Let M and N be topological spaces,let G be a group,and let τ:G×M→M be a proper free action of G.In this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair(G,τ) that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S~n to R~n.In the cases where M is a finite pathwise-connected CWcomplex,G is a finite,non-trivial Abelian group,τ is a proper free cellular action,and N is either R~2 or a compact surface without boundary different from S~2 and RP~2,we give an algebraic criterion involving braid groups to decide whether a free homotopy class β∈[M,N] has the Borsuk-Ulam property.As an application of this criterion,we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group Zn.In terms of the orient ability of the orbit space Mof M by the action τ,the value of n modulo 4 and a certain algebraic condition involving the first homology group of M,we are able to determine if the single homotopy class of maps from M to R~2 possesses the Borsuk-Ulam property with respect to(Z,τ).Finally,we give some examples of surfaces on which the symmetric group acts,and for these cases,we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R~2.