[Paper Review] Twisted longitudinal index theorem for foliations and wrong way functoriality

Paulo Carrillo Rouse, Bai‐Ling Wang|arXiv (Cornell University)|May 20, 2010

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper establishes a twisted longitudinal index theorem for foliations using deformation quantization and Connes' tangent groupoid, generalizing the Connes-Skandalis index theorem to twisted K-theory. It constructs a wrong way functorial pushforward in twisted K-theory for smooth maps to foliated manifolds, proving functoriality and recovering known results in the untwisted and manifold cases.

ABSTRACT

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai-Melrose-Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map $f:W\longrightarrow M/F$ and a twisting $σ$ on the holonomy groupoid $M/F$, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes-Skandalis when the twisting is trivial and of Carey-Wang for manifolds.

Motivation & Objective

• To extend the Connes-Skandalis longitudinal index theorem to the setting of twisted K-theory on foliated manifolds.
• To define an analytic index morphism in twisted K-theory for Lie groupoids with a twisting via deformation quantization and the tangent groupoid construction.
• To establish a wrong way functoriality (pushforward) in twisted K-theory for smooth maps from manifolds to foliated spaces with twistings.
• To generalize the results of Connes-Skandalis (untwisted case) and Carey-Wang (twisted manifold case) to the foliated setting.
• To lay the foundation for a twisted assembly map in geometric K-homology for foliations, with connections to S¹-equivariant indices.

Proposed method

• Uses the Connes tangent groupoid construction to define the analytic index morphism for twisted Lie groupoids via deformation to the normal cone.
• Applies geometric deformation quantization techniques to define a twisted analytic index from the K-theory of the Lie algebroid to the K-theory of the twisted groupoid C*-algebra.
• Constructs the pushforward map in twisted K-theory by factorizing smooth maps through submersions and using the pushforward in twisted K-theory for submersions.
• Employs the Thom isomorphism and Bott periodicity in topological K-theory to relate the topological index to the K-theory of the foliation C*-algebra.
• Proves functoriality of the pushforward via composition of factorizations and uses the functoriality of the pushforward in [7] (Carey-Wang).
• Relies on the Morita equivalence of the C*-algebra of a foliated open neighborhood with continuous functions vanishing at infinity on a transversal.

Experimental results

Research questions

• RQ1 How can the Connes-Skandalis longitudinal index theorem be generalized to the case of twisted K-theory on foliated manifolds?
• RQ2 What is the correct formulation of the analytic index morphism in twisted K-theory for Lie groupoids equipped with a twisting?
• RQ3 How can a wrong way functoriality (pushforward) be defined and proven in twisted K-theory for smooth maps from manifolds to foliated spaces?
• RQ4 In what way does the twisted index construction recover known results in the untwisted case (Connes-Skandalis) and the twisted manifold case (Carey-Wang)?
• RQ5 What is the role of the tangent groupoid and deformation quantization in constructing the twisted index and pushforward maps?

Key findings

• The paper constructs a twisted analytic index morphism for Lie groupoids with a twisting σ, generalizing the untwisted case and recovering the Connes-Skandalis index when σ is trivial.
• The twisted longitudinal index theorem is proven: the analytic index equals the topological index in twisted K-theory for foliated manifolds.
• The pushforward map in twisted K-theory is defined for any smooth map f:W→M/F with a twisting σ on the holonomy groupoid, using factorization through submersions.
• The pushforward is proven to be functorial: (f∘g)_! = f_! ∘ g_! for composable smooth maps.
• When σ is trivial and f is K-oriented, the construction recovers the pushforward of Connes-Skandalis.
• When the foliation has a single leaf (i.e., M/F is a manifold), the construction reduces to the pushforward in twisted K-theory defined by Carey-Wang.