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.