C. 3i Working Paper Alfred P. Sloan School of Management a Unified Approach to Some Combinatorial Lemmas in Topology 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 lenmas in topology. \Je demonstrate two new dual lemmas on the n-dimensional cube, and use a Generc.lized S ^erner Lemma to prove a gener lization of the Knaster-Kuratowski-y.azurkiewicz 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, -^the existence of stationary points on the simplex^ Introduction 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 Borsak-Ulam Theorem. We report the interrelationships between these results, the Brouwer Fixed Point Theorem, the Borsak-Ulam Theorem, the existence of stationary points on the simplex, and two new theorems on the simplex relating to the Generalized Covering Theorem. In Section I, we give constructive proofs of the Sperner Lemma, Scarf's Dual Sperner Lemma, and a Generalized Sperner Lemma, We show that the Generalized Sperner Lemma leads to simple proofs of the Sperner Lemma and the Dual Sperner Lemma. We show how each of these results is related to Brouwer 's Fixed Point Theorem. Section II introduces a Generalized Covering Theorem that generalizes the Knaster-Kuratowski-Mazurkiewicz Lemma [12]. The Generalized Covering Lemma is used to provide a proof of the existence of stationary points on the simplex, as well as two new results on the simplex. Section III deals with combinatorial lemmas on the cube that relate to Brouwer 's Theorem. Two new dual lemmas on the cube are introduced and given constructive proofs. We also give constructive proofs of Gale's Hex Theorem and Kuhn's Strong Cubical Lemma. It is shown that the Hex Theorem and one of the new lemmas are equivalent. Finally, the interrelationship between these results and Brouwer 's Theorem are exposited.