# 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.