Variable Dimension Complexes Part II: A Unified Approach to Some Combinatorial Lemmas in Topology

Part II of this study uses the path-following theory of labelled V-complexes developed in Part I to provide constructive algorithmic proofs of a variety of combinatorial lemmas in topology. We demonstrate two new dual lemmas on the n-dimensional cube, and use a generalized Sperner lemma to prove a generalization of the Knaster–Kuratowski–Mazurkiewicz covering lemma on the simplex. We also show that Tucker's lemma can be derived directly from the Borsuk–Ulam theorem. We report the interrelationships between these results, Brouwer's fixed point theorem, and the existence of stationary points on the simplex.