The holonomy Lie $\infty$-groupoid of a singular foliation I

This paper constructs a finite-dimensional Kan simplicial manifold that integrates the universal Lie $\infty$-algebroid of a singular foliation admitting a geometric resolution, utilizing a recursive application of bi-submersions to generalize the Androulidakis-Skandalis holonomy groupoid to a higher Lie groupoid setting.

The Gist

Here is an explanation of the paper "The holonomy Lie 8-groupoid of a singular foliation I" using simple language, analogies, and metaphors.

The Big Picture: Mapping a Chaotic City

Imagine you are trying to map a city. In a normal city (a regular foliation), the streets are perfectly laid out in a grid. Every block is the same size, and you can easily walk from one to the next. Mathematicians have had a perfect map for these cities for a long time.

But what if the city is chaotic?

• Some streets are wide boulevards.
• Some are narrow alleyways.
• Some areas are just a single point (a dead end).
• Some areas are huge open parks.

This is a Singular Foliation. It's a way of dividing a space into "leaves" (the streets/areas), but these leaves can change size and shape unpredictably. The problem is: How do you build a map (a groupoid) for this chaotic city that captures all the rules of movement, even when the streets change size?

For decades, mathematicians have struggled to build a single, smooth, finite-dimensional map for these chaotic cities. Previous attempts either resulted in maps that were infinitely complex (too big to hold in your head) or only worked for specific, simple cases.

The Breakthrough: The "Bi-Submersion" Tool

The authors of this paper, Camille Laurent-Gengoux and Ruben Louis, have built a new kind of map. They call it a Holonomy Lie 8-groupoid.

To understand how they did it, imagine they are using a special tool called a Bi-Submersion.

The Analogy of the Bi-Submersion:
Think of a bi-submersion as a universal translator or a bridge between two different versions of the city.

• Imagine you have a map of the city drawn on a flat sheet of paper ($M$).
• You also have a 3D model of the city ($W$).
• A bi-submersion is a way to project the 3D model onto the paper and back again, ensuring that every "street" in the 3D model corresponds perfectly to a "street" on the paper, even if the 3D model is twisted or folded.

The genius of the paper is that they don't just use one bridge. They build a Tower of Bridges.

The Construction: Building a Tower of Maps

The authors realized that to understand the chaos of the singular foliation, you can't just look at the ground level. You need to look at the "vertical" structure of the chaos.

1. Level 0 (The Ground): This is the city itself ($M$).
2. Level 1 (The First Bridge): They build a bridge ($K_1$) that connects points in the city. This bridge is like the "Holonomy Groupoid" from previous work, but it's just the first step. It tells you how to get from point A to point B.
3. Level 2 (The Bridge Between Bridges): Now, imagine you have two different ways to get from A to B. How do you know if those two paths are "equivalent"? They build a second layer ($K_2$) that connects the bridges themselves. It's like a "meta-map" that organizes the relationships between the first bridges.
4. Level 3, 4, ... 8 (The Infinite Ladder): They keep going up. $K_3$ connects the relationships of Level 2, and so on.

They call this a "Lie 8-groupoid" not because there are exactly 8 levels, but because "8" in math often implies "infinity" or "all levels." They are building an infinite tower of maps where each level organizes the one below it.

The Secret Ingredient: The "Geometric Resolution"

The paper has a condition: The chaotic city must admit a "Geometric Resolution."

The Metaphor:
Imagine the chaotic city is a tangled ball of yarn.

• A Geometric Resolution is like having a set of tools that can untangle the yarn layer by layer.
• First, you pull out the big loops (Level 1).
• Then you pull out the smaller knots (Level 2).
• Then the tiny tangles (Level 3).

If you can untangle the city into a finite number of layers (a geometric resolution), the authors' method works. If the city is infinitely tangled in a way that can't be untangled, their method doesn't apply yet.

The Result: A Finite, Perfect Map

The most exciting part of the paper is that their tower of maps is Finite-Dimensional.

• Old Problem: Previous attempts to map these cities resulted in maps that were infinitely large (infinite-dimensional). It was like trying to draw a map on a piece of paper that keeps growing forever.
• New Solution: The authors prove that if you have a geometric resolution, you can build a map where every level ($K_1, K_2, K_3...$) is a specific, finite size. You can actually calculate the dimensions!

They also introduce a concept called a "Para-Lie 8-groupoid."

• The Analogy: Usually, a perfect map follows strict rules (like a rigid grid). Their map is slightly more flexible. It follows the rules of a grid mostly, but allows for a little bit of "wiggle room" in how the pieces fit together (specifically regarding "degeneracy" or repeating points). It's a "quasi-perfect" map that is good enough to do the math but flexible enough to handle the chaos.

Why Does This Matter?

1. It Solves a 60-Year-Old Question: Mathematicians have been asking, "Can we always find a Lie groupoid (a smooth map) for these singular foliations?" The answer was "No" for the old, rigid definitions. This paper says, "Yes, but you have to look at it through a higher-dimensional lens (the 8-groupoid)."
2. It's Practical: Because the map is finite-dimensional, it can be used in physics and engineering. Singular foliations appear in:
   • Poisson Geometry: Describing how particles move in magnetic fields.
   • Control Theory: Steering robots or spacecraft where the rules of movement change depending on where you are.
   • Quantum Mechanics: Understanding the "symmetries" of complex systems.

Summary

Imagine you have a messy, shifting puzzle.

• Old Math: Tried to force the puzzle into a rigid box. It didn't fit, or the box became infinitely big.
• This Paper: Builds a tower of flexible bridges (the bi-submersion tower) that connects the puzzle pieces.
• The Magic: If the puzzle can be untangled layer by layer (geometric resolution), this tower is a manageable, finite size.
• The Outcome: We now have a complete, finite-dimensional "map" (the Holonomy Lie 8-groupoid) for a whole new class of chaotic mathematical spaces.

The authors have essentially given us a new way to navigate the chaos of the universe, turning an infinite mess into a structured, finite tower of understanding.

Technical Summary

Here is a detailed technical summary of the paper "The holonomy Lie 8-groupoid of a singular foliation I" by Camille Laurent-Gengoux and Ruben Louis.

1. Problem Statement

The paper addresses a fundamental open question in differential geometry regarding singular foliations. While regular foliations (where leaves have constant dimension) are well-understood via Lie groupoids and Lie algebroids, singular foliations (where leaf dimensions vary) pose significant integration challenges.

• Question 0.1: Given a singular foliation $\mathcal{F}$ on a manifold $M$, does there exist a Lie groupoid $G \to M$ whose associated Lie algebroid integrates $\mathcal{F}$?
  • Current Status: The answer is generally "no" for arbitrary singular foliations, even when the number of local generators is globally bounded. Obstructions exist, and standard integration methods often fail to produce finite-dimensional objects or require infinite-dimensional structures (e.g., Fréchet manifolds).
• Question 0.2: Can a singular foliation be integrated into a Lie $\infty$-groupoid (a Kan simplicial manifold) that is finite-dimensional? Specifically, can one construct a natural, unique (up to homotopy) finite-dimensional object whose differentiation yields the universal Lie $\infty$-algebroid of $\mathcal{F}$?

2. Methodology

The authors propose a novel integration method that avoids infinite-dimensional analysis and gauge-fixing conditions used in previous works (e.g., Širaň and Ševera). Instead, they rely on bi-submersions, a concept from non-commutative geometry introduced by Androulidakis and Skandalis.

Key Methodological Steps:

1. Geometric Resolutions: The construction assumes the singular foliation $\mathcal{F}$ admits a geometric resolution. This is an anchored complex of vector bundles $(E_\bullet, d, \rho)$ such that the sequence of sections is exact:
   $$\dots \to \Gamma(E_{-2}) \xrightarrow{d} \Gamma(E_{-1}) \xrightarrow{\rho} \mathcal{F} \to 0$$
   This class includes real analytic and holomorphic singular foliations.
2. Bi-submersions: The authors utilize bi-submersions $W$ between foliated manifolds $(M, \mathcal{F})$ and $(N, \mathcal{G})$. These are manifolds equipped with two submersions $p, q$ satisfying specific compatibility conditions with the foliations. They serve as local charts for the holonomy groupoid.
3. Recursive Construction (Bi-submersion Towers):
   • The authors construct a sequence of manifolds $K_0, K_1, K_2, \dots$ recursively.
   • $K_0 = M$.
   • $K_1$ is constructed as a "holonomy atlas" (a disjoint union of bi-submersions) derived from the geometric resolution.
   • Higher levels $K_n$ are constructed as equivalences between the previous level $K_{n-1}$ and the horn spaces $\Lambda^n_\ell K$ (fiber products representing partial simplices).
   • This process forms a bi-submersion tower, where each step resolves the "vertical" singular foliation (bi-vertical vector fields) of the previous level.
4. Para-simplicial Structure: The resulting structure is not a strict simplicial manifold because the degeneracy maps ($s_i$) do not satisfy all simplicial identities ($s_i s_j = s_{j+1} s_i$). The authors define this weaker structure as a para-simplicial manifold. However, it satisfies the Kan condition (horn filling), which is sufficient for the definition of a Lie $\infty$-groupoid in this context.

3. Key Contributions

• Finite-Dimensional Integration: The paper constructs a finite-dimensional object (a Kan simplicial manifold) integrating any singular foliation admitting a geometric resolution. This contrasts with previous infinite-dimensional approaches.
• Holonomy Para-Lie $\infty$-groupoid: The authors define a new structure called the holonomy para-Lie $\infty$-groupoid. It generalizes the Androulidakis-Skandalis holonomy groupoid (which is the 1-truncation of this structure) to higher orders.
• Universal Lie $\infty$-algebroid Integration: The construction proves that the differentiation of this para-Lie $\infty$-groupoid yields the universal Lie $\infty$-algebroid of the singular foliation, which is unique up to homotopy.
• Dimensional Control: The authors provide an explicit formula for the dimension of the $k$-th level $K_k$:
  $$\dim K_k = \dim M + \sum_{i=1}^k \binom{k}{i} \text{rk}(E_{-i})$$
  where $E_{-i}$ are the vector bundles in the geometric resolution.
• New Theory of Bi-submersions: The paper significantly expands the theory of bi-submersions, introducing concepts like bi-transversals, total relations, and bi-vertical singular foliations. It proves that equivalent bi-submersions can be reduced to manifolds of constant dimension (Theorem D.17).

4. Main Results

Theorem 2.3: Every singular foliation $\mathcal{F}$ on a manifold $M$ that admits a geometric resolution admits a holonomy para-Lie $\infty$-groupoid $K_\bullet(\mathcal{F})$.

• Properties:
  1. $K_0 = M$.
  2. $K_1$ is a holonomy bi-submersion atlas.
  3. For $k \ge 1$, $K_k$ is a manifold where all connected components have the same dimension given by the formula above.
  4. The tangent complex of $K_\bullet$ restricted to $M$ recovers the geometric resolution of $\mathcal{F}$.
  5. The 1-truncation $K_1 / \sim$ is the classical Androulidakis-Skandalis holonomy groupoid.
  6. If the geometric resolution has length $n$ (i.e., $E_{-k} = 0$ for $k > n$), then for $k > n$, the horn projections are surjective local diffeomorphisms, and the structure stabilizes.

5. Significance

• Resolution of the Integration Problem: This work provides a positive answer to Question 0.2 for a broad class of singular foliations (those with geometric resolutions), establishing that they can be integrated into finite-dimensional higher groupoids.
• Bridge between Geometry and Homotopy Theory: It connects the geometric theory of singular foliations (via bi-submersions) with homotopical algebra (Lie $\infty$-algebroids and Kan complexes), offering a concrete geometric realization of abstract algebraic structures.
• Foundation for Future Work: By introducing the "para-simplicial" framework and the "bi-submersion tower," the authors lay the groundwork for studying higher symmetries, deformation quantization, and the cohomology of singular foliations in a finite-dimensional setting.
• Generalization of Known Results: It recovers the classical holonomy groupoid for regular foliations and the Debord groupoid for projective foliations as specific truncations or special cases of this higher structure.

In summary, the paper successfully constructs a finite-dimensional, higher-categorical geometric object that integrates singular foliations, overcoming previous limitations regarding dimensionality and the existence of Lie groupoid structures.