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 ℤ_n. In terms of the orientability of the orbit space M_τ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of M_τ, we are able to determine if the single homotopy class of maps from M to ℝ² 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 ℝ².