A Künneth Formula for the Embedded Homology

Abstract

Hypergraph is an important model for complex networks. A hypergraph can be regarded as a virtual simplicial complex with some faces missing and it is the key hub to connect the simplicial complex in topology and graph in combinatorics. The embedded homology groups of hypergraphs are new developments in mathematics in recent years, and the embedded homology groups of hypergraphs can reflect the topological and geometric characteristics of complex network which can not be reflected by the associated simplicial complex of hypergraphs. Kunneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In this paper, we prove that the infimum chain complex of tensor products of free R-modules generated by hypergraphs is isomorphic to the tensor product of their respective infimum chain complexes, and give an analogues of Kunneth formula for hypergraphs by classical algebraic Kunneth formula based on the embedded homology groups of hypergraphs, which provides a theoretical basis for further study of cohomology theory of hypergraphs. In fact, the Kunneth formula here can be extended to the Kunneth formula of embedded homology of graded abelian groups of chain complexes, which can be used to extend the Kunneth formula for digraphs with coefficients in a field.