Construction of multiply fixed n-valued maps

Abstract

A self-map, either single-valued or multi-valued, is multiply fixed if every map homotopic to it has at least two fixed points. Let p:X˜→X be a finite covering space, of degree n, of a connected finite polyhedron, and let f:X→X be a map. We lift f to an n-valued map ϕp,f:X˜⊸X˜ and prove that it is multiply fixed if the Nielsen number of f is greater than or equal to two. We obtain a formula for the Nielsen number of ϕp,f in terms of the Nielsen number of f, the induced fundamental group homomorphism of f, and the monodromy action of the covering space. We describe specific constructions of the n-valued maps ϕp,f on graphs, orientable double covers, handlebodies, free G-spaces and nilmanifolds.