Cohomological Aspects of Hypergraphs

By a &-graph we will mean a collection of k-element subsets of some fixed set V. A A:-graph can be regarded as a (k l)-chain on 2y, the simplicial complex of all subsets of V, over the coefficient group Z/2 , the additive group of integers modulo 2. The induced group structure on the {k 1)chains leads to natural definitions of the coboundary Ô of a chain, the cochain complex of C = {Ck, ô) and the usual cohomology groups Hk(C; Z/2). In particular, it is possible to construct what could be called "higher-order" coboundary operators <5''>, where <5<'> increases dimension by /' (rather than just 1). In this paper we will develop various properties of these J'1', and in particular, compute the corresponding cohomology groups for 2V over Z/2. It turns out that these groups depend in a rather subtle way on the arithmetic properties of i.