Hodge’s harmonic p-sets and Pontrjagin classes

This paper shows how Hodge’s theory of harmonic p-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author’s definition of flag vectors for hypergraphs, and other objects constructed out of vertices and cells. It is not widely known that Hodge developed a discrete version of his theory of harmonic differential forms. The purpose of this paper is to show how this theory leads to a new approach to the problem of finding a purely combinatorial description of the Pontrjagin classes of a compact manifold. Gelfand and MacPherson [3] have already solved this problem. Their solution uses oriented matroids, which are complicated combinatorial objects that can encode ‘differential structure’. The Hodge approach, if successful, is likely to be simpler and more explicit. It may also lead to fresh insight into characteristic classes and related invariants. In this paragraph, M will be a compact differential manifold of dimension n, equipped with a Riemannian metric. Now suppose that η is any closed p-form on M , perhaps representing a Pontrjagin class of M . Hodge’s theory of harmonic forms can now be applied. It yields a unique p-form ηh, that is equivalent to η, and harmonic for the given metric. Now let A be any p-cell of M . The number ηA = ∫