Complexes of not i-connected graphs

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by Vassiliev [38, 39, 41]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n−2)! spheres of dimension 2n−5. This answers a question raised by Vassiliev in connection with knot invariants. For this case the S_n-action on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n−2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n−3)-connected graphs we provide a formula for the generating function of the Euler characteristic