Extensions of Sperner and Tucker's lemma for manifolds

TL;DR

This paper extends Sperner and Tucker's lemmas, which are combinatorial equivalents of Brouwer and Borsuk-Ulam theorems, to a broad class of manifolds beyond discs and spheres, expanding their applicability.

Contribution

It generalizes classical combinatorial lemmas to include a wide range of manifolds, not just discs and spheres, broadening their theoretical scope.

Findings

Lemmas hold for various manifolds with or without boundary

Classical combinatorial lemmas are applicable to more complex topological spaces

Potential new applications in topological combinatorics and related fields

Abstract

The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk - Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.