Equivariant indices of Spin$^c$-Dirac operators for proper moment maps

Some remarks on equivariant elliptic operators and their invariants

In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and the equivariant index. Many researchers have previously considered these invariants before. What makes this work different is that we are evaluating integer-valued indices corresponding to multiplicities of group representations, and our eta invariant is a number dependent on the entire group at once. Moreover, the techniques of proof and formulas obtained ar…

Projective elliptic genera and elliptic pseudodifferential genera

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in Mathai-Melrose-Singer (2006) without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of …

真余紧G-流形的等变Lefschetz不动点定理

The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds

设给定一个离散群G，一个余紧真G-流形M，以及M的一个G-自映射f。然后，我们引入了f的等变Lefschetz类和局部等变Lefschetz类，前者是全局定义的胞腔链复形，后者是局部定义的不动点.我们证明了等变Lefschetz不动点定理，它说，这两个类同意。作为特例，我们证明了一个等变Poincare-Hopf定理，利用等变向量场的零点计算了泛等变欧拉示性数，并得到了一个轨道型Lefschetz不动点定理.最后，我们证明了泛等变欧拉示性数的一个实现定理。

极作用的等变LS-范畴

The Equivariant LS-Category of Polar Actions

我们将给出任意真作用量的等变Lusternik-Schnirelmann范畴在轨道类型分层意义下的一个下界，以及真极作用量的等变Lusternik-Schnirelmann范畴在广义外尔群意义下的一个上界.作为应用，我们重新证明了Singhof的一个定理，该定理决定了U（n）和SU（n）的经典Lusternik-Schnirelmann范畴.

真作用的等变解析挠率

Equivariant analytic torsion for proper actions

本文为局部紧群在黎曼流形上的真等距作用构造了等变Ray-Singer解析挠率，证明了其收敛性、度量独立性与消失性等性质，并推广了先前仅适用于有限群、紧群或紧共轭类的相关结果。

Equivariant holonomy of U(1)-bundles

We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show that the equivariant holonomy can be used to obtain results about equivariant prequantization and anomaly cancellation.

矩阵群具有紧商的等变Poincaré对偶

An equivariant Poincaré duality for proper cocompact actions by matrix groups

本文证明了线性李群在具有紧商的$G$-spin$^c$流形上真等距作用时，基于有限维$G$-向量丛定义的$G$-等变$K$-理论与基于Baum-Douglas几何模型定义的$G$-等变$K$-同调之间的Poincaré对偶定理。

Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types

In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers…

Equivariant discretizations of diffusions and harmonic functions of bounded growth

For covering spaces and properly discontinuous actions with compatible diffusion operators, we discuss Lyons-Sullivan discretizations of the associated diffusions and harmonic functions of bounded growth.

An equivariant orbifold index for proper actions

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal comp…

An equivariant index for proper actions II: properties and applications

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, we show how it is related to the analytic assembly map from the Baum-Connes conjecture, and an index used by Mathai and Zhan…

A fixed point theorem on noncompact manifolds

We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results…

A topological approach to indices of geometric operators on manifolds with fibered boundaries

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define $K$-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete $Φ$ or edge metrics, can be regarded as the index pairing over these $K$-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.

An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index

Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index $\operatorname{index}_g(D)$ was defined for an element $g \in G$, in term…

正标量曲率与适当作用下的等变 Callias 型指标定理

Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

本文研究局部紧群适当作用流形上具有一致正标量曲率的完备不变度量存在性，利用等变 Callias 型指标定理给出障碍，并在连通李群情形下推广 Lawson-Yau 构造消除障碍。

Coarse geometry and Callias quantisation

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group $C^*$-algebra instead of its reduced …

旋转等变Minkowski赋值

Rotation equivariant Minkowski valuations

投影体算子是连续赋值，它对应于欧几里得空间Rn中的每个凸体及其投影体，平移不变，旋转等变.也是众所周知的是，Rn将Rn中的多面体集合映射到自身。我们证明了，R2是唯一的非平凡的运营商与这些属性。