Holonomy and singular foliations

A manifold partitioned into immersed manifolds – also called leaves – is a foliated manifold. Foliations arise while solving differential equations in different fields of mathematics, including mathematical physics and control theory dealing with the behaviour of dynamical systems.

However, while well-behaved 'regular' foliations have been extensively studied, the majority of foliations are pathological. These singular foliations, which appear on manifolds as a submodule of compactly supported vector fields, were the focus of the project NCGSF (Noncommutative geometry for singular foliations).

Researchers formulated the Baum-Connes (BC) conjecture for any given singular foliation. This conjectural generalisation of the Atiyah-Singer theorem states that purely topological objects coincide with purely analytical ones. Its proof was possible thanks to the construction of a so-called pathological holonomy groupoid.

The holonomy groupoid is a mathematical structure that keeps track of foliations' symmetries. This was the key to development of the analytical part of the BC conjecture. Specifically, the researchers introduce the notion of holonomy transformation, an equivalence class of diffeomorphisms.

For the formulation of the geometrical part, the researchers deployed the LeGall-Tu model. But first, it was necessary to define the conditions ensuring the longitudinal smoothness of the holonomy groupoid carefully. Only then was it possible to formulate the model of the normal form of a regularly foliated manifold around a compact leaf.

The methodology followed within NCGSF has been documented in a series of three publications in international peer-reviewed journals. It builds on previous results of the researchers who refined this further with the construction of the holonomy groupoid of any singular foliation.