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 ver tices. 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 ques tion 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 compl exes of not 2-connected X-hypergraphs we provide a formula for the generati ng function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for s ome other cases. In particular, we show that the complex of not (n - 2)-con nected 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 t he generating function of the Euler characteristic. (C) 1998 Elsevier Scien ce Ltd. All rights reserved.