New cohomological invariants of foliations
Georges Habib, Ken Richardson
TL;DR
The paper develops a cohomological theory for foliations via antibasic forms, defining $H_a^{k}(M,\mathcal{F},g)$ and proving a Hodge-type decomposition and spectral theory for the antibasic Laplacian $\Delta_a$ in the Riemannian setting. It shows antibasic cohomology is a foliation invariant and, in many cases, combines with basic cohomology to recover overall de Rham information, e.g., $H^{k}(M)\cong H_b^{k}(M,\mathcal{F})\oplus H_a^{k}(M,\mathcal{F})$ when the normal bundle is involutive and the mean curvature is basic. The analysis yields elliptic regularity, discreteness of the spectrum, and a robust Hodge decomposition for antibasic forms, along with foliated homotopy invariance and a suite of topological restrictions on antibasic Betti numbers. The work also provides explicit computations in several examples, including Riemannian and non-Riemannian foliations, to illustrate when antibasic invariants yield additional information beyond basic or ordinary cohomology. Overall, the results furnish new foliation invariants and analytic tools for understanding transverse geometry and its interaction with ambient topology.
Abstract
Given a smooth foliation on a closed manifold, basic forms are differential forms that can be expressed locally in terms of the transverse variables. The space of basic forms yields a differential complex, because the exterior derivative fixes this set. The basic cohomology is the cohomology of this complex, and this has been studied extensively. Given a Riemannian metric, the adjoint of the exterior derivative maps the orthogonal complement of the basic forms to itself, and we call the resulting cohomology the "antibasic cohomology". Although these groups are defined using the metric, the dimensions of the antibasic cohomology groups are invariant under diffeomorphism and metric changes. If the underlying foliation is Riemannian, the groups are foliated homotopy invariants that are independent of basic cohomology and ordinary cohomology of the manifold. For this class of foliations we use the codifferential on antibasic forms to obtain the corresponding Laplace operator, develop its analytic properties, and prove a Hodge theorem. We then find some topological and geometric properties that impose restrictions on the antibasic Betti numbers.