Cubes and cubical chains and cochains in combinatorial topology

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 independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-Schnirelmann theorems which is both conceptual and elementary. It is based on adaptation of Serre's definition of products of singular cubical cochains to discrete setting. The main results are new purely combinatorial "cubical lemmas". This approach also clarifies the cubical versions of Sperner lemma of Kuhn and Ky Fan. In particular, Ky Fan's lemma can be understood as a natural strengthening of Lebesgue or Kuhn's results under a transversality assumption. The exposition does not assume any knowledge of algebraic topology.