The Complex of 2-Connected Graphs

John Shareshian

Mathematics Department
Cal Tech

Monday, February 9, 1998
4:00 PM

Room 102, Bradley Hall

Let S(n) be a simplex with one vertex for each pair of distinct elements from the set [n]={1,...,n}. The faces of S(n) correspond naturally to graphs on vertex set [n]. Let NTC(n) be the subcomplex of S(n) whose faces correspond to graphs which are not 2-connected and let TC(n) be the quotient space S(n)/NTC(n). The complexes TC(n) arise in Vassiliev's study of knot invariants. The natural action of the symmetric group S_n on [n] determines an action on TC(n) which in turn gives a linear representation of S_n on the homology of TC(n). In this talk I will show how to determine a basis for the unique nontrivial reduced homology group of TC(n). This result answers a question of Vassiliev and provides insight into the connection between the linear representation mentioned above and representations of S_n on other combinatorial and algebraic objects.