Fulton–MacPherson Configuration Space

• Fulton–MacPherson configuration space is a smooth, compactified version of configuration spaces, achieved by resolving point collisions through iterated blow-ups.
• The construction methodically blows up diagonals to create a stratified manifold with corners, enabling precise analysis of nested collision data.
• Applications span algebraic geometry, topology, operad theory, and mathematical physics, offering robust tools for studying invariants and moduli spaces.

Fulton-MacPherson Configuration Space

The Fulton–MacPherson configuration space is a smooth, compactification of the configuration space of pairwise distinct points on a manifold, constructed by an explicit resolution of the singular strata corresponding to point collisions via an iterated blow-up of diagonals. This construction, originally introduced by Fulton and MacPherson in 1994, produces a manifold (or scheme) with corners whose boundary stratifies configurations according to nested collision data. The resulting spaces have become fundamental objects in algebraic geometry, topology, operad theory, and mathematical physics, possessing rich geometric, combinatorial, and homotopical structure.

1. Construction via Iterated Blow-Ups and Alternate Models

Let $M$ be a smooth $n$-manifold and $k \geq 1$ an integer. The ordered configuration space is

$$\mathrm{Conf}_k(M) = M^k \setminus \bigcup_{i < j} \Delta_{ij}$$

where $\Delta_{ij} \subset M^k$ is the “small diagonal” where the $i$-th and $j$-th points coincide. The Fulton–MacPherson compactification, denoted $\mathrm{FM}_M(k)$ or $M[k]$, is constructed by systematically blowing up all diagonals $\Delta_I$ for $I \subset \{1,\dots,k\}$, $|I|\geq2$, in increasing order of intersection (i.e., increasing cardinality) (Idrissi, 2016, Vrecica et al., 2008, Mok, 21 Mar 2025, Gallardo et al., 2021, Massarenti, 2016, Joshi, 2010).

This yields a smooth manifold with corners, equipped with a proper map

$\pi: \mathrm{FM}_M(k) \rightarrow M^k,$

which is a diffeomorphism over the locus of pairwise-distinct points, and whose boundary decomposes into smooth divisors indexed by collision types.

Equivalently, the compactification can be described as the closure of the embedding

$$\mathrm{Conf}_k(M) \hookrightarrow M^k \times \prod_{|I| \geq 2} S(N\Delta_I),$$

where each factor records the normalized direction of approach of points in $I$, and the closure systematically “remembers” all relative directions and scales of collision (Idrissi, 2016, Vrecica et al., 2008).

In algebraic geometry, for a smooth variety $X$ over an algebraically closed field, the construction proceeds identically, with the blow-ups performed in the category of schemes, and is known to be compatible under étale or smooth pullbacks (Mok, 21 Mar 2025).

2. Geometry, Boundary Stratification, and Local Structure

The manifold $\mathrm{FM}_M(k)$ inherits a canonical stratification by locally closed submanifolds (or locally closed subschemes) corresponding to sequences of nested collisions:

• Each stratum corresponds to a flag of subsets $I_1 \subsetneq I_2 \subsetneq \dots \subsetneq I_r$ with $|I_j| \geq 2$.
• The codimension-$r$ stratum parametrizes configurations where the clusters $I_j$ successively collapse.

The boundary divisors in $\mathrm{FM}_M(k)$ correspond to the exceptional divisors produced from blowing up the diagonals $\Delta_I$; these divisors intersect transversely and their intersection patterns are governed by "nestings" of the index sets, e.g., intersections are nonempty precisely when the corresponding subsets are nested or disjoint (Mok, 21 Mar 2025, Massarenti, 2016, Gallardo et al., 2021, Idrissi, 2016).

Local charts near a boundary stratum indexed by a nested sequence of subsets are provided by:

• Center-of-mass coordinates for each cluster,
• For each colliding block, polar (direction and scale) coordinates in the normal bundle,
• Iterated such charts for nested collisions (Vrecica et al., 2008, Gootjes-Dreesbach, 15 Apr 2025).

This structure imbues $\mathrm{FM}_M(k)$ with a manifold-with-corners or simple-normal-crossing log-structure (Mok, 21 Mar 2025, Vrecica et al., 2008).

3. Operadic and Homotopical Structure

When $M = \mathbb{R}^n$, the collection $\{\mathrm{FM}_n(k)\}_{k \geq 1}$ forms a topological operad known as the Fulton–MacPherson or “FM” operad, which admits an explicit composition law: $$\circ_i : \mathrm{FM}_n(k) \times \mathrm{FM}_n(l) \to \mathrm{FM}_n(k + l - 1)$$ by inserting the second configuration infinitesimally at the $i$-th point of the first and rescaling (Idrissi, 2016, Salvatore, 2019, Tourtchine, 2012, Salvatore, 2019, Bottman, 2021). This structure is strictly compatible with the manifolds-with-corners and all boundary strata, making FM_n a topological (and piecewise semi-algebraic) $E_n$-operad, weakly equivalent to the little $n$-disks operad.

The FM operad admits variants for filtered manifolds (with weighted blow-ups, yielding generalized configuration spaces compatible with filtrations (Gootjes-Dreesbach, 15 Apr 2025)), graphical/wonderful compactifications (for configuration spaces with prescribed collision graphs (Khoroshkin et al., 25 Sep 2025)), and weighted/partial diagonals (recovering moduli of weighted pointed rational curves (Gallardo et al., 2016)).

For manifolds $M$ with trivialized tangent bundle (framed case), the spaces $\mathrm{FM}_M(k)$ form a right $FM_n$-module via insertion of configurations in fibers identified with $\mathbb{R}^n$ (Idrissi, 2016, Tourtchine, 2012).

4. Algebraic and Homotopical Models, Formality, and Invariants

The real homotopy type of $\mathrm{FM}_M(k)$ for a simply connected closed manifold $M$ (with $\dim M \geq 4$) is fully determined by the real homotopy type of $M$ itself. Explicit CDGA models are provided by the Lambrechts–Stanley model: $$G(A, k) = [A^{\otimes k} \otimes H^*(\mathrm{FM}_n(k))]/\text{(Arnold relations)}$$ with differential

$d\,\omega_{ij} = \iota_{ij}(\Delta_A),$

where $A$ is a Poincaré duality CDGA model for $M$ and $H^*(\mathrm{FM}_n(k))$ satisfies Arnold’s relations (Idrissi, 2016). This algebra models the real cohomology and is equipped with Hopf comodule and operad/module structures compatible with that of FM_n and the little disks operad.

Furthermore, by constructing explicit graph complexes and a zigzag of quasi-isomorphisms, one identifies the PA forms on $\mathrm{FM}_M(k)$ with combinatorial graph complexes decorated by $H^*(M)$, extending Kontsevich’s formality and factorization homology results to closed manifolds (Campos et al., 2016, Idrissi, 2016).

For configuration spaces in filtered manifolds or graphical configuration spaces, analogous combinatorial or weighted models provide algebraic and differential invariants relevant from both topological and representation-theoretic perspectives (Gootjes-Dreesbach, 15 Apr 2025, Khoroshkin et al., 25 Sep 2025).

5. Logarithmic, Weighted, and Degenerate Compactifications

Recent developments generalize the Fulton–MacPherson construction via logarithmic geometry and weighted blow-ups:

• The logarithmic Fulton–MacPherson configuration space provides a canonical, functorial log-smooth compactification over fs log schemes, compatible with tropical and Artin fan structures (Mok, 21 Mar 2025). In this setting, the boundary and its stratification acquire a "tropical" combinatorial interpretation, and log smooth degenerations admit explicit "degeneration formulas" expressing special fibers as unions of birational modifications of products of log FM spaces.
• Weighted compactifications accommodate filtered or jet spaces by performing "weighted" blow-ups along diagonals, where the weighting corresponds to the order of vanishing or underlying Lie filtration. The outcome is a smooth manifold with corners generalizing the usual FM space while capturing higher-order collision data (Gootjes-Dreesbach, 15 Apr 2025).

For graphical configuration spaces associated to a finite graph $\Gamma$, one constructs FM-type compactifications by blowing up, in building set order, the "tube diagonals" corresponding to connected subgraphs, yielding contractads that extend the FM operad and are deeply connected to the combinatorics of the graph (Khoroshkin et al., 25 Sep 2025).

6. Applications, Automorphisms, and Birational Geometry

Fulton–MacPherson spaces serve pivotal roles in the study of moduli spaces, embedding calculus, quantum knot invariants, and the homotopy theory of operads:

• For $X[n]$ with $X$ a smooth projective variety, the boundary divisors stratify by “collision type” and are indexed by nonempty proper subsets. Stratification codifies in terms of rooted trees or weighted trees, matching the combinatorics of moduli of stable curves and certain GIT quotients (Gallardo et al., 2016, Gallardo et al., 2021).
• The automorphism group of $X[n]$ exhibits rigidity: for $n \neq 2$ or $\dim X \geq 2$, $$\operatorname{Aut}^0(X[n]) \cong \operatorname{Aut}^0(X)$$ (Massarenti, 2016). For curves and suitable generalizations, the full automorphism group includes permutation of indices and diagonal copies of automorphisms of the factor.
• Birational geometry of $X[n]$ is intricate; for large enough $n$, these spaces fail to be Mori dream spaces, with effective cones that are non-polyhedral and Cox rings that are not finitely generated (Gallardo et al., 2021).
• In manifold calculus and embedding theory, FM compactifications serve as models for stages of the Taylor tower, and their algebraic invariants (operad/module structures) quantify and govern the embedding spaces' homotopy types (Tourtchine, 2012).
• Applications to knot theory, link invariants, and the study of chord diagrams and cyclohedra (via special configurations such as on $S^1$) showcase the flexibility and reach of these spaces (Vrecica et al., 2008).

7. Cellular and Simplicial Structures, Combinatorics, and Further Developments

FM configuration spaces and their operads admit rich combinatorial models:

• FM_2 and higher FM_n have explicit regular CW or simplicial decompositions, with cells indexed by nested trees (or metatrees with cacti labels and edge colorings), making operad compositions strictly cellular (Salvatore, 2019, Bottman, 2021). This combinatorics is deeply linked to the structure of E₂-operads and the Deligne conjecture.
• Simplicial models compatible with the FM operad have been constructed, which allow for strict cellularity in chain-level structures, relevant for chain-Hochschild complexes and symplectic field theory (Bottman, 2021).
• Enumeration and generating function techniques for cell-counts yield algebraic and recursion-theoretic formulas, with connections to tree enumeration and cacti operads (Salvatore, 2019).
• Extensions include weighted, logarithmic, and graphical FM compactifications, each preserving a version of the manifold-with-corners or log-smooth structure and encoding detailed algebraic or combinatorial data of collision types (Mok, 21 Mar 2025, Gootjes-Dreesbach, 15 Apr 2025, Khoroshkin et al., 25 Sep 2025).

The wealth and flexibility of the Fulton–MacPherson construction underlie its ubiquity and centrality across topology, algebraic geometry, moduli theory, and deformation analysis. Its operad-theoretic avatars and compatibility with modern techniques in log-geometry, factorization homology, and higher category theory continue to fuel active research and applications.