A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula

Foliated group actions and cyclic cohomology

We prove a cyclic Lefschetz formula for foliations. To this end, we define a notion of equivariant cyclic cohomology and show that its expected pairing with K-theory is well defined. This enables to associate to any invariant transverse current on a compact foliated manifold, a Lefschetz formula for leafwise preserving isometries.

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. We associate with such a current an equivariant cyclic cohomology class of Connes' C*-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points. As special cases, we recover a number of classical results, and since we may use any closed Haefliger current on the foliation, we get new and very interesting formulae.

非紧不动点集的 Lefschetz 不动点公式

A Lefschetz fixed-point formula for noncompact fixed-point sets

本文针对完备 Riemannian 流形上的等距映射，提出了适用于非紧不动点集的等变指标定理（Lefschetz 不动点公式），通过引入局部化泛函与渐近局部算子代数，证明了已有指标为其特例并得到了正数量曲率的障碍。

Equivariant eta forms and equivariant differential $K$-theory

In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary dimensional fibers.Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilize…

关于Γ-恰当流形上的更高莱夫舍茨公式

Higher Lefschetz formulas on Γ-proper manifolds

本文提出了一种在Γ-恰当流形上应用热核技术，推导出狄拉克算子的更高莱夫舍茨公式的几何方法。

等变代数K-理论与导出完备化 III：应用

Equivariant Algebraic K-Theory and Derived completions III: Applications

本文利用前序论文的导出完备化定理，在一般情形下建立了线性代数群作用下等变G-理论与等变同伦K-理论的Riemann-Roch定理，证明其适用于环面簇与球面簇，还得到Lefschetz-Riemann-Roch定理及相关计算谱序列。

Equivariant noncommutative motives

Given a finite group G, we develop a theory of G-equivariant noncommutative motives. This theory provides a well-adapted framework for the study of G-schemes, Picard groups of schemes, G-algebras, 2-cocycles, equivariant algebraic K-theory, orbifold cohomology theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin's motivic theory. As a first application, we extend Panin's computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfec…

本质半群交叉积的Baum-Connes装配映射

A Baum-Connes assembly map for essential semigroup crossed products

构造了可分离C*-代数上逆半群Fell丛的等变E-理论和Baum-Connes装配映射，推广了先前工作，并建立了相关函子性质。

$\mathrm{G}$-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem

We develop a version of $\mathrm{G}$-theory for an $\mathbb{F}_1$-algebra (i.e., the $\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\mathrm{K}$-theory for finite pointed groups with our $\mathrm{G}$-theory. We compute the $\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypot…

Higher traces, noncommutative motives, and the categorified Chern character

We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (inf…

Bivariant $K$-theory with $R/Z$-coefficients and rho classes of unitary representations

We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $Γ$ be a group. We define a $Γ$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $Γ$ acts as the unit element in $KK^Γ_{\mathbb{R}}(A,A)$. We show that free and proper $Γ$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $Γ$ is torsion free and …