Nielsen coincidence theory applied to Borsuk–Ulam geometric problems

This work uses Nielsen coincidence theory to discuss solutions for the geometric Borsuk–Ulam question. It considers triples (X,τ;Y) where X and Y are topological spaces and τ is a free involution on X, (X,τ;Y) satisfies the Borsuk–Ulam theorem if for any continuous map f:X→Y there exists a point x∈X such that f(x)=f(τ(x)). Borsuk–Ulam coincidence classes are defined and a notion of essentiality is defined. The classical Borsuk–Ulam theorem and a version for maps between spheres are proved using this approach.