  With a view toward studying the homotopy type of spaces of Boolean formulae,
we introduce a simplicial complex, called the theta complex, associated to any
hypergraph, which is the Alexander dual of the more well-known independence
complex. In particular, the set of satisfiable formulae in k-conjunctive normal
form with less than or equal to n variables has the homotopy type of
Theta(Cube(n,n-k)), where Cube(n,n-k) is a hypergraph associated to the
(n-k)-skeleton of an n-cube. We make partial progress in calculating the
homotopy type of theta for these cubical hypergraphs, and we also give
calculations and examples for other hypergraphs as well. Indeed studying the
theta complex of hypergraphs is an interesting problem in its own right.
