COHOMOLOGICAL ASPECTS OF HYPERGRAPHS

By a k-graph we will mean a collection of k-element subsets of some fixed set V. A k-graph can be regarded as a (k - 1)-chain on 2V, 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 delta of a chain, the cochain CoMpleX of C = {C(k), delta} and the usual cohomology groups H(k)(C; Z/2). In particular, it is possible to construct what could be called "higher-order" coboundary operators delta(i) , where delta(i) increases dimension by i (rather than just 1). In this paper we will develop various properties of these delta(i), and in particular, compute the corresponding cohomology groups for 2V over Z/2. It tums out that these groups depend in a rather subtle way on the arithmetic properties of i.