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The present paper is a continuation of author's paper arXiv:1909.00940[math.AT] devoted to the lemmas of Alexander and Sperner, but is independentfrom it. We begin by a step back from Alexander and Sperner to Lebesgue work onthe invariance of the dimension. In contrast with almost everybody else,Lebesgue worked with cubes rather than with simplices. His methods weredeveloped by Hurewicz and Lusternik-Schnirelmann and then forgotten. In thepresent paper these methods are recast in the language of cubical chains andcochains.  After this, we present a new approach to Lebesgue and Lusternik-Schnirelmanntheorems which is both conceptual and elementary. It is based on adaptation ofSerre's definition of products of singular cubical cochains to discretesetting. The main results are new purely combinatorial "cubical lemmas". Thisapproach also clarifies the cubical versions of Sperner lemma of Kuhn and KyFan. In particular, Ky Fan's lemma can be understood as a natural strengtheningof Lebesgue or Kuhn's results under a transversality assumption.  The exposition does not assume any knowledge of algebraic topology.

Cubes and cubical chains and cochains in combinatorial topology