A cell decomposition of the Fulton MacPherson operad

Abstract

We construct a small regular cellular decomposition of the Fulton MacPherson operad 𝐹𝑀 2 that is compatible with the operad composition. The cells are indexed by trees with edges of two colors and vertices labeled by cells of the cacti operad. We compute the generating functions counting the cells that are algebraic.

Koszul duality for topological E n ‐operads

“…such that: • 0 𝑖 = • 𝑖 , and if 𝑦 ∈ 𝐹 𝑉 (𝐼) or 𝑦 ′ ∈ 𝐹 𝑉 (𝐽) is in the image of some composition map, then 𝑦• 𝑠 𝑖 𝑦 ′ is in the image of the corresponding map, compare [33]. Now let 𝑇 ′ be an 𝐼-labelled tree, and 𝑇 ′′ a 𝐽-labelled tree.…”

“…This inclusion extends to a collar neighbourhood of the face in a way that preserves the face structure. In other words, there is an inclusion $$\begin{equation*} \circ _i^{\bullet }: [0,\infty ] \times F_V(I) \times F_V(J) \rightarrow F_V(I \cup _i J) \end{equation*}$$such that: $$\circ _i^0 = \circ _i$$, and if $$y \in F_V(I)$$ or $$y^{\prime } \in F_V(J)$$ is in the image of some composition map, then $$y \circ _i^s y^{\prime }$$ is in the image of the corresponding map, compare [33]. Now let $$T^{\prime }$$ be an $$I$$‐labelled tree, and $$T^{\prime \prime }$$ a $$J$$‐labelled tree.…”

The Fulton–MacPherson operad and the $W$-construction

“…is an embedding, and its image is the closure of the stratum of F n (k) indexed by T . See Section 6 of [8].…”

“…An explicit isomorphism for n = 1 is described in [1]. The isomorphism for n = 2 can be constructed explicitly using the machinery of [8], as we explain later. An important application of the main theorem is the construction of an operadic cellular decomposition of the Fulton-MacPherson operad F 2 described in [8].…”

“…The isomorphism for n = 2 can be constructed explicitly using the machinery of [8], as we explain later. An important application of the main theorem is the construction of an operadic cellular decomposition of the Fulton-MacPherson operad F 2 described in [8]. The key insight of the proof is that F n (k) is a manifold with corners, and W F n (k) can be identified to a standard fattening of F n (k), that is homeomorphic to it.…”