The Borsuk-Ulam theorem for maps into a surface

TL;DR

This paper characterizes when the Borsuk-Ulam property holds for maps from a CW-complex with an involution into a surface, using braid groups and fundamental group classifications, especially for surfaces without boundary.

Contribution

It provides a classification of triples (X, t, S) satisfying the Borsuk-Ulam property based on fundamental group isomorphisms and involution properties, extending previous results to broader surface cases.

Findings

Classifies triples (X, t, S) where the Borsuk-Ulam property holds.

Shows the property generally fails unless specific fundamental group conditions are met.

Provides necessary and sufficient conditions involving homomorphisms for orientable cases.

Abstract

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, we formulate this property in terms of a relation in the 2-string braid group B_2(S) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples (X, t, S) for which the Borsuk-Ulam property holds. We also consider various cases where X is not necessarily a surface without boundary, but has the property that \pi_1(X/t) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S^2 and the real projective plane RP^2, then we show that the Borsuk-Ulam property does not hold for (X, t, S) unless either \pi_1(X/t) is isomorphic to \pi_1(RP^2), or \pi_1(X/t) is isomorphic to the fundamental group of a compact, connected non-orientable surface of genus 2 or 3 and S is orientable. In the latter case, the veracity of the Borsuk-Ulam property depends further on the choice of involution t; we give a necessary and sufficient condition for it to hold in terms of the surjective homomorphism \pi_1(X/t)-->Z_2 induced by the double covering X-->X/t. The cases S=S^2,RP^2 are treated separately.