Categories of acyclic graphs and automorphisms of free groups
Gunnar Carlsson, Woong Kook · 1996
The acyclic graph structures on a given set of n vertices defines a simplicial complex, called an acyclic graph complex. We compute the homology of this complex as a module over the symmetric group $\Sigma\sb{n}$ on n letters. Combinatorially it is a shellable complex, a special kind of Cohen-Macaulay complex. The shellability implies that the reduced homology of this complex is concentrated in the top dimension. There is a permutation basis for the homology over the symmetric group on $n-1$ letters. We analyze a transposition action on this basis and complete the description of the homology as a $\Sigma\sb{n}$-module. Possible applications of the acyclic graph complexes to outer automorphisms of free groups are outlined.