Generalizations of Tucker-Fan-Shashkin lemmas

TL;DR

This paper extends classical combinatorial lemmas related to the Borsuk-Ulam theorem to a broader class of manifolds called BUT-manifolds, using generalized theorems and manifold doubling techniques.

Contribution

It introduces new generalizations of Tucker-Fan-Shashkin lemmas for BUT-manifolds, expanding their applicability beyond spheres.

Findings

Generalized lemmas for BUT-manifolds

Proofs based on a generalized odd mapping theorem

A lemma on doubling manifolds with boundaries

Abstract

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.