Index theory and non-commutative geometry on foliated manifolds

The index of operators on foliated bundles

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geometry.

Index of transverse Dirac operator and cohomotopy Seiberg-Witten invariant for codimension $4$ Riemannian foliation

For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic cohomological group is of zero dimension, we can give an estimate for the index of transversal Dirac operator of a foliated spin structure. Furthermore, under the condition that $H^\pm_b(M)=1$, we show the vanish…

叶状仿射与射影结构

Foliated affine and projective structures

本文形式化复流形上沿全纯叶状结构叶片的仿射与射影结构，给出奇点局部正规形式和紧致情形下的指标公式，并证明一般型正则叶状结构不承认叶状射影结构，最终分类了紧复曲面上的此类结构。

Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds

We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.

Pseudodifferential operators on manifolds with foliated boundaries

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…

可扩大叶状结构与单值广群：无限覆盖

Enlargeable Foliations and the Monodromy Groupoid: Infinite Covers

研究非紧可扩大 spin 叶状结构的叶状 Rosenberg 指数，利用相对指标定理及紧算子 C*-代数与 C 的 KK 等价克服无限维困难，证明该指数非零。

Perturbations of basic Dirac operators on Riemannian foliations

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.

李群胚 Riemann-Roch-Hirzebruch 定理及应用

Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

本文引入李代数胚指标理论并研究其 Dolbeault 算子，结合正则叶状流形上的 Connes 指标理论建立广义 Riemann-Roch 定理，并提出 Kodaira 消灭定理以通过拓扑信息验证正线丛判据。

Non-commutative analytic torsion form on the transformation groupoid convolution algebra

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our construction is well defined under the weaker assumption of positive Novikov-Shubin invariant. We prove that this torsion form appears in a transgression formula, from which a non-commutative Riamannian-Roch-Grothendieck index formula follows.

Twisted higher index theory on good orbifolds and fractional quantum numbers

The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in the Hall effect on the hyperbolic plane.

叶状结构的高阶相对指标定理

Higher relative index theorems for foliations

本文将 Gromov 和 Lawson 的相对指标定理推广至非紧流形叶状结构的叶向 Dirac 算子，利用 Connes-Chern 特征的高阶项证明了相对拓扑指标与解析指标相等，并以此证明叶向正标量曲率度量空间具有无穷多连通分支。

Chern characters in equivariant basic cohomology

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 usef…

Longitudinal b-operators, Blups and Index theorems

Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. We further use the blup groupoids as the one above, and in particular its functoriality pr…

Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a Toeplitz operator with Hölder continuous symbol. The index formula gives an analytic formula for the degree of a Hölder continuous mapping from the boundary of a strictly pseudo-convex domain.

Analytic formulas for topological degree of non-smooth mappings: the even-dimensional case

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölder continuous vector bundle. The index formula gives an analytic formula for the degree of a Hölder continuous mapping between even-dimensional manifolds. The paper is an independent continuation of the paper "Analytic formulas for topological degree of non-smooth mappings: the od…

Projective Dirac Operators, Twisted K-Theory and Local Index Formula

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the A-hat genus …

An index theorem for families invariant with respect to a bundle of Lie groups

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singer type formula. We also study traces on the corresponding algebras of pseudodifferential operators and obtain a local index formula for such families of invariant operators, using the Fedosov product. For topologically non-trivial bundles we have to use methods of non-commutative…