| 2 | Free Cyclic Actions on Surfaces and the Borsuk-Ulam Theorem显示文摘Let M and N be topological spaces,let G be a group,and letτ: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,τis 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 actionτof the finite cyclic group Zn.In terms of the orient ability of the orbit space Mof M by the actionτ,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_(n),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). | Daciberg Lima GONCALVES John GUASCHI Vinicius Casteluber LAASS | 2022 | Acta Mathematica Sinica,English Series2022，38，10： | 0 |
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