The Eilenberg-Zilber Theorem via Discrete Vector Fields∗

The Eilenberg-Zilber (EZ) Theorem is basic in Algebraic Topology. It connects the chain complex C∗(A×B) of the product of two simplicial sets to the tensor product C∗(A)⊗ C∗(B), allowing one to compute H∗(A×B) from H∗(A) and H∗(B). Initially proved in 1953, it is stated and used in almost every elementary textbook of Algebraic Topology. Sixty-five years after the initial EZ paper, we present this result via a totally new proof, based on the relatively recent tool called Discrete Vector Field (DVF, Robin Forman, 1998). The various available EZ proofs are not so trivial; the present proof on the contrary is intuitive and gives more information. The key point is the notion of s-path, a 2-dimensional description of every simplex of the standard simplicial decomposition of ∆p×∆q, p and q arbitrary, which allows us to use the same DVF for the homotopical and homological EZ theorems. The right notion of morphism between cellular chain complexes provided with discrete vector fields is given; its definition is really amazing and is here an essential tool. Besides a new understanding of the EZ theorem, we obtain much better algorithms for the machine implementation of the EZ theorem, automatically avoiding the countless degenerate simplices produced by the RubioMorace formula for the EZ-homotopy. Other striking applications will be the subject of other papers.