COMPLEXES OF NOT i-CONNECTED GRAPHS

Abstract

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. 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 one of the questions raised by Vassiliev [V3] in connection with knot invariants. For this case the S naction 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.1 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 2 There is Z 3 -torsion of rank 8. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 3 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5, 7. 4 There is Z 3 -torsion of rank 35. No Z p -torsion for p = 2, 5, 7. 5 There is Z 3 -torsion of rank 56. No Z p -torsion for p = 2, 5, 7.

Topology of matching, chessboard, and general bounded degree graph complexes

“…The matching complex also appeared in a 1999 paper of Babson, Björner, Linusson, Shareshian and Welker [2] on the graph complex consisting of graphs on node set [n] which are not k-connected. A graph is said to be k-connected if removal of any j nodes, where j = 0, 1, .…”

“…When k = 1, 2, the complex arises in connection with Vassiliev knot invariants [47,48]. It was observed in [2] that when k = n−3, the not k-connected graph complex is the Alexander dual of the matching complex M n .…”

“…The integral homology groups of M n , for n ≤ 12, are shown in Table 6.1 which first appeared in [2]. They were computed using a computer program of Heckenbach which was later updated by Dumas, Heckenbach, Saunders and Welker [22].…”

“…Graph complexes have provided an important link between combinatorics and algebra, topology and geometry; see eg., [29,11,47,54,2,48,37]. Here we consider the simplicial complexes associated with the collection of subgraphs of a (di-, multi-, hyper-)graph G whose node degrees are bounded from above.…”

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“…There is another recent motivation for the computation of the rational homology of the complete graph matching complex n , ensuing from work of Vassiliev, which is discussed in [4]. In particular, Table 3 of that reference lists homology calculations ofH i ( m,n ; k) for small values of i, char(k) and Theorem 1.3 (or the results of [6,18]) accurately predict all of the non-torsion data which occurs in this table.…”