A cell decomposition of the Fulton MacPherson operad

Abstract

We construct a regular cellular decomposition of the Fulton MacPherson operad F M 2 that is compatible with the operad composition. The cells are indexed by trees with edges of two colors and vertices labelled by cells of the cacti operad. This answers positively a conjecture stated by Kontsevich and Soibelman in 2000. We compute the generating functions counting the cells, that are algebraic.