Free Cyclic Actions on Surfaces and the Borsuk—Ulam Theorem

Abstract

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 $${\mathbb{S}^n}$$ to ℝn. In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either ℝ2 or a compact surface without boundary different from $${\mathbb{S}^2}$$ and ℝℙ2, 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 orientability of the orbit space Mτ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of Mτ, we are able to determine if the single homotopy class of maps from M to ℝ2 possesses the Borsuk—Ulam property with respect to (ℤn, 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 ℝ2.