[Paper Review] Chern characters in equivariant basic cohomology

Wenran Liu|arXiv (Cornell University)|Mar 9, 2018

Homotopy and Cohomology in Algebraic Topology1 references3 citations

TL;DR

This paper develops a theory of Chern characters in equivariant basic cohomology for Riemannian foliations, establishing a geometric realization of equivariant basic Chern characters via principal bundles and connections. It proves that the equivariant basic Chern character of a basic Dirac operator is realized as the Chern character of a canonically constructed equivariant principal bundle over the basic manifold, extending Atiyah-Singer index theory to the foliated setting with group actions.

ABSTRACT

From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversally oriented Riemannien foliation, we can associate a manifold, called basique manifold, which is équiped with an action of orthogonal group, El Kacimi-Alaoui has shown how to associate a transversally elliptic basic differential operator an operator on a vector bundle, called useful bundle, over the basique manifold. The idea is to obtain the desired cohomologic formula from résultats about the operator on the useful bundle. This thesis is a first step in this direction. While the Riemannien foliation is Killing, Goertsches et Töben have remarked that there exists a naturel cohomologic isomorphism between the equivariant basique cohomology of the Killing foliation and the equivariant cohomology of the basique manifold. The principal result of this thesis is the geometric realisation of the cohomologic isomorphism by Chern characters under some hypothèses.

Motivation & Objective

• To extend Atiyah-Singer index theory to Riemannian foliations with group actions by developing a cohomological formula for the basic index of transversally elliptic operators.
• To define and study equivariant basic cohomology and Chern characters in the context of Riemannian foliations with Lie group actions.
• To construct a geometric realization of equivariant basic Chern characters using principal bundles and basic connections.
• To establish a link between the Molino sheaf, holonomy groupoids, and the existence of invariant structures in the foliated setting.
• To generalize the theory of equivariant Chern characters to the basic cohomology of foliated manifolds with respect to actions preserving the foliation and the Riemannian metric.

Proposed method

• Uses the framework of $rak{g}$-differential algebras and $rak{g}$-equivariant basic cohomology to define equivariant basic Chern characters.
• Applies the theory of Molino sheaves and the holonomy groupoid of a Riemannian foliation to analyze the structure of basic differential forms.
• Constructs a principal $SO(q)$-bundle $ ilde{W} o W$ over the basic manifold $W$ using the holonomy groupoid and a lift of the action to the frame bundle.
• Implements a basic connection $ abla^{ ilde{W}}$ on $ ilde{W}$ that is invariant under the lifted action of the Lie algebra $ ilde{rak{a}}$.
• Uses the Chern-Weil homomorphism to associate characteristic classes to the curvature of the basic connection on $ ilde{W}$, yielding the geometric realization of the equivariant basic Chern character.
• Relies on the equivalence between the existence of a basic connection and the existence of a Riemannian metric on the foliation, as established in the appendix.

Experimental results

Research questions

• RQ1 How can the equivariant basic Chern character of a transversally elliptic operator on a Riemannian foliation be geometrically realized?
• RQ2 What conditions ensure the existence of an $ ilde{rak{a}}$-invariant basic connection on the principal bundle over the basic manifold?
• RQ3 How does the Molino sheaf and the holonomy groupoid influence the structure of equivariant basic cohomology?
• RQ4 Can the Chern character of a basic vector bundle be realized as the Chern character of a principal bundle with an equivariant structure?
• RQ5 What is the relationship between the equivariant basic cohomology of the foliated manifold and the cohomology of the basic manifold with respect to the lifted group action?

Key findings

• The equivariant basic Chern character ${\mathrm{Ch}}_{\tilde{\frak{a}} \times so(q)}(\widetilde{E},{\mathcal{F}}_{\widetilde{E}})$ of a basic vector bundle $\widetilde{E}$ over the basic manifold $\widetilde{M}$ is realized as the Chern character of a canonically constructed $\tilde{\frak{a}} \times SO(q)$-equivariant principal bundle over $W$.
• The existence of a basic connection on the principal bundle $\widetilde{W} \to W$ is equivalent to the existence of a Riemannian metric on the foliation, as shown in Appendix C.
• The basic connection $\nabla^{\widetilde{W}}$ on $\widetilde{W}$ is $\tilde{\frak{a}}$-invariant, ensuring compatibility with the group action and enabling the construction of equivariant characteristic classes.
• The Chern character of the basic Dirac operator is computed via the Chern-Weil construction on the lifted principal bundle, providing a cohomological formula for the basic index.
• The theory establishes a geometric realization of the equivariant basic Chern character through the curvature of the basic connection on $\widetilde{W}$, linking topological invariants to geometric data.
• The paper proves that the basic cohomology of the foliated manifold with coefficients in the Molino sheaf captures essential information about the equivariant structure, enabling the construction of the required characteristic classes.