Combinatorial analogs of Brouwer's fixed-point theorem on a bounded polyhedron

Abstract

In this paper, we present a combinatorial theorem on a bounded polyhedron for an unrestricted integer labelling of a triangulation of the polyhedron, which can be interpreted as an extension of the Generalized Sperner lemma. When the labelling function is dual-proper, this theorem specializes to a second theorem on the polyhedron, that is, an extension of Scarf's dual Sperner lemma. These results are shown to be analogs of Brouwer's fixed-point theorem on a polyhedron, and are shown to generalize other combinatorial theorems on bounded polyhedra as well. The paper also contains a pseudomanifold construction for a polyhedron and its dual that may be of interest to researchers in triangulations based on primal and dual polyhedra.