Coincidence Sets of Coincidence Producing Maps

Abstract

AbstractA theorem by H. Robbins shows that every closed and non-empty subset of the unit ball Bn in Euclidean n-space is the fixed point set of a self map of Bn. This result is extended to coincidence producing maps of Bn, where a map ƒ:X → Y is coincidence producing (or universal) if it has a coincidence with every map g:X → Y. The main result implies that if ƒ:Bn, Sn - 1 → Bn, Sn - 1 is coincidence producing and A⊂Bn closed and nonempty, then there exist a map ƒ': Bn, Sn - 1 → Bn, Sn - 1 and a map g: Bn → Bn such that ƒ' | Sn - 1 is homotopic to ƒ | Sn-1 and A is the coincidence set of ƒ' and g.