Complexes of trees and nested set complexes

Eva Maria Feichtner

Vol. 227 (2006), No. 2, 271–286

DOI: 10.2140/pjm.2006.227.271

Abstract

We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees T_n and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context.

Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another generalization, more natural in the context of nested set complexes.

Keywords

complexes of phylogenetic trees, nested set complexes, cohomology bases

Mathematical Subject Classification 2000

Primary: 05E25

Secondary: 57Q05

Milestones

Received: 5 February 2005

Accepted: 22 June 2005

Published: 1 October 2006