Cohomological aspects of hypergraphs

Abstract:

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 ${2^V}$, the simplicial complex of all subsets of $V$, over the coefficient group $\mathbb {Z}/2$, 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 ${H^k}(C;\mathbb {Z}/2)$. 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 ${2^V}$ over $\mathbb {Z}/2$. It turns out that these groups depend in a rather subtle way on the arithmetic properties of $i$.