[Paper Review] Cech-De Rham theory for leaf spaces of foliations

Marius Crainic, Ieke Moerdijk|ArXiv.org|Dec 10, 2000

Homotopy and Cohomology in Algebraic Topology15 references4 citations

TL;DR

This paper introduces a Cech-De Rham model for the cohomology of leaf spaces of foliations, providing a geometric, non-Hausdorff-free alternative to traditional models like classifying spaces or holonomy groupoids. By constructing characteristic classes directly on the leaf space using explicit geometric methods, it establishes the Bott vanishing theorem and Poincaré duality at the level of the leaf space cohomology, with applications to universal characteristic classes and Ruelle-Sullivan currents.

ABSTRACT

The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical constructions which are immediate extensions of the standard constructions for manifolds. In particular we rediscover (and explain) the Thurston formula and the Bott formulas for cocycles on diffeomorphism groups. We also use the Cech-De Rham model to explicitly describe the relation between the cohomology of the classifying space, the basic cohomology, and the foliated cohomology of foliations.

Motivation & Objective

• To overcome the limitations of existing models for leaf space cohomology, such as non-Hausdorff structures and loss of smoothness in classifying spaces.
• To provide a geometric, direct construction of characteristic classes in the cohomology of the leaf space $M/\mathcal{F}$, rather than in $H^*(M)$.
• To establish Poincaré duality and the Bott vanishing theorem at the level of the leaf space cohomology using a new Cech-De Rham model.
• To relate the new cohomology model to basic cohomology, foliated cohomology, and the cohomology of holonomy groupoids via spectral sequences and Van Est-type maps.
• To give an explicit geometric realization of universal characteristic classes via the Haefliger groupoid, recovering formulas such as Thurston’s and Bott’s.

Proposed method

• The paper constructs a Cech-De Rham double complex using a transversal basis $\mathcal{U}$ of transversal sections to define cochains on the leaf space.
• It defines a cohomology theory $H^*(M/\mathcal{F})$ via the total cohomology of the Cech-De Rham complex, avoiding reliance on holonomy groupoids or classifying spaces.
• The Van Est map $\Phi: H^s(M/\mathcal{F}; \Omega^t_{\text{bas}}) \to H^s(\mathcal{F}; \Lambda^t \nu)$ is used to relate the Cech-De Rham cohomology to the foliated cohomology of the foliation.
• Integration along the leaves is defined via a map $\int_{\mathcal{F}}: H^p_c(\mathcal{F}) \to H^{p-q}_c(M/\mathcal{F}; \Omega^0_{\text{bas}})$, dually related to the Van Est map.
• Spectral sequences are employed to relate the Cech-De Rham model to basic cohomology and foliated cohomology, with $E^{s,t}_1 = H^s(M/\mathcal{F}; \Omega^t_{\text{bas}})$ converging to $H^{s+t}(M/\mathcal{F})$.
• The model is shown to be quasi-isomorphic to the cohomology of the holonomy groupoid via étale groupoid techniques, proving equivalence with established models.

Experimental results

Research questions

• RQ1 Can a geometric, non-Hausdorff-free cohomology theory for leaf spaces be constructed that supports direct constructions of characteristic classes?
• RQ2 Does the Cech-De Rham model recover known results like the Bott vanishing theorem and Poincaré duality at the level of the leaf space cohomology?
• RQ3 How are the cohomology of the leaf space, basic cohomology, and foliated cohomology related via spectral sequences and natural maps?
• RQ4 Can the universal characteristic classes of the Haefliger groupoid be explicitly constructed geometrically using this model?
• RQ5 What is the role of integration along leaves and transverse measures in realizing the Ruelle-Sullivan current?

Key findings

• The Cech-De Rham model provides a cohomology theory for the leaf space $M/\mathcal{F}$ that is geometric, avoids non-Hausdorff issues, and supports direct constructions of characteristic classes.
• The Van Est map $\Phi$ induces an isomorphism between $H^s(M/\mathcal{F}; \Omega^t_{\text{bas}})$ and $H^s(\mathcal{F}; \Lambda^t \nu)$ in degrees $s \leq k$ when the holonomy covers of leaves are $k$-connected.
• The integration along the leaves map $\int_{\mathcal{F}}$ is dual to the Van Est map and provides a geometric realization of the Ruelle-Sullivan current as a degree $p$ current on $M$.
• The Cech-De Rham cohomology with compact supports satisfies a duality with the standard cohomology, which recovers the Poincaré duality of [10] as a natural extension of standard arguments from manifolds.
• The model recovers the Thurston and Bott formulas for cocycles on diffeomorphism groups by providing a geometric construction of universal characteristic classes via the Haefliger groupoid.
• Characteristic classes of transversal principal bundles over $(M, \mathcal{F})$ are shown to lift from $H^*(M)$ to $H^*(M/\mathcal{F})$ via the Cech-De Rham model, with all standard relations preserved.