KK-theory of C*-categories and the analytic assembly map
Paul D. Mitchener
我们定义KK理论谱与C*-范畴，并在此水平上的卡斯帕罗夫产品的某些情况下看。这台机器是用来给一个描述的分析组装地图作为一个自然的地图的光谱。
We define KK-theory spectra associated to C*-categories and look at certain instances of the Kasparov product at this level. This machinery is used to give a description of the analytic assembly map as a natural map of spectra.
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Algebraic $K$-theory Spectra and Factorisations of Analytic Assembly Maps
在这篇文章中，我们使用现有的机器来定义连接$K$理论光谱拓扑环状。作为特例，得到了离散环的代数$K$ -理论和Banach范畴的解析$K$ -理论.作为一个应用程序，我们展示了如何解析组装地图中的诺维科夫和鲍姆-康纳斯结构可以分解成复合材料的组装地图类似于那些出现在代数$K$ -理论和地图来完成某些拓扑ringoids到巴拿赫类别。这些factorisations证明了使用现有的装配图的特征，以及我们的统一的图片代数和解析$K$理论。
The Baum-Connes Conjecture via Localisation of Categories
我们重新定义Baum-Connes装配映射使用单纯逼近在等变Kasparov范畴。这种新的解释是理想的研究函子的性质，并给出类似的组装地图的所有等变同调论，而不仅仅是为K-理论的交叉产品。我们扩展了许多已知的技术证明鲍姆-康纳斯猜想，这个更一般的设置。
A model for the assembly map of bordism-invariant functors
本文通过研究稳定范畴、埃尔米特范畴和庞加莱范畴的oplax余极限，构造了配边不变函子装配映射的范畴模型，并利用该模型将Shaneson分裂推广到带扭曲的情形，同时推广了Kirstein-Kremer的近期工作。
C*-categories, Groupoid Actions, Equivariant KK-theory, and the Baum-Connes Conjecture
本文给出了具有系数的Baum-Connes组装映射的一个刻划。所需要的技术工具是C*-范畴的K-理论和群胚世界中的等变KK-理论。
We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on homological algebra in triangulated categories, is compatible with the previously studied deformation picture of the assembly map, and allows us to define an assembly map with arbitrary coefficients for these quantum groups.
Splitting the relative assembly map, Nil-terms and involutions
证明了当系数是具有群作用的加法范畴时，代数K理论的有限子群族到虚循环子群族的相对Farrell-Jones组装映射是分裂内射的.这推广了Bartels关于系数环的一个结果.我们给出了一个明确的描述的相对项。这使我们能够证明，如果我们在正则环中取系数，它有理地为零。此外，如果G的I型无限虚循环子群是可定向的，则通过取对偶模的对合，它被认为是Z[Z/2]-模，扩张模，特别是它的所有Tate上同调群为零。例如，后一个条件对于无挠双曲群是满足的。
The Analytical Assembly Map and Index Theory
本文研究了Baum-Connes猜想中出现的解析集合映射的指数理论解释。它的一般形式可以用卡斯帕罗夫的等变KK理论来构造。在特殊情况下的torsionfree群域简化为通常的K-同调的分类空间BG的G和它是经常使用的，在这种情况下，分析组装地图是通过分配给一个运营商的等变指数。我们给出了这个陈述的精确公式并证明了它。
In this note we set a configuration space description of the equivariant connective K-homology groups with coefficients in a unital C*-algebra for proper actions. Over this model we define a connective assembly map and prove that in this setting is possible to recover the analytic assembly map.
Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model
我们构造了Higson和Roe的解析外科群的几何模拟，用于值在群代数的巴拿赫代数完备化中的群的自由作用的装配映射.我们证明了几何定义群与解析外科群类似，在集合映射下可满足一个六项正合列，并讨论了几何群的定义域映射.特别是，给定两个有限维酉表示相同的秩，我们定义了一个映射的精神$η$型不变量从几何组（相对于组装的完全群$C^*$ -代数）的实数。
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly map explicitly in terms of locally finite homology theory.
Paschke duality and assembly maps
本文研究了 Paschke 变换在 $G$-空间上的等价性，并将其应用于比较 Davis-Lück 组装映射与 Kasparov 分析组装映射。
Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map i…
Towards the Baum-Connes' Analytical Assembly Map for the Actions of Discrete Quantum Groups
给定一个离散量子群的作用（在Van Daele，Kustermans和Effros-Ruan的意义下）${\cal A}$在$C^*$ -代数${\cal C}$上，满足类似于经典离散群$Γ$在某个空间上的真$Γ$ -紧作用的正则性假设，我们能够构造出规范映射$μ^i_r$（\$μ_i $ respectively) ($ i=0，1 $) from the $ { \cal A} $-equivariant K-homology groups $ KK_i^{ \cal A}（{ \cal C}，C|）$ to the K-theory groups $ K_i（\hat{{\cal A}_r}）$ ($ K_i（\hat{\cal A}）$ respectively), where $ \hat{{\cal A}_r} $ and $ \hat{\cal A} $ stand for the quantum analogues of the reduced and full group $ C^*$-代数（c.f. [11]，[6]）。我们遵循经典Baum-Connes映射的构造步骤，尽管在量子群的上下文中，必须认真考虑不变权重的非平凡模性质（以及相关的事实，即对极的平方不是单位的），这使得猜测和证明相关希尔伯特模结构的正确定义有点棘手。
The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions
本文研究了带系数的巴乌-康纳斯装配映射e*和米斯钦科-卡斯帕罗夫装配映射μ*在群扩张中的单射性、满射性和同构性，证明了在特定条件下这些映射的性质可以传递到扩展群。
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geo…
Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of Segal as developed by the author in arXiv:2111.14313 [math.KT]. The resulting new definition of the analytic index makes sense under much weaker continuity assumptions than the Atiyah-Singer one and can be easily adjusted to families of operators in fibers of a Hilbert bundle. We prove the correctness of the new definition and show that it agrees with the A…
The signature package on Witt spaces, II. Higher signatures
本文是“Witt空间上的特征包，I.《分类目录》作者相同。在第一部分中，我们调查，通过一个参数化的建设，在分层维特伪流形上的签名算子的正则性，证明，特别是，可以定义一个K-同调签名类。我们还建立了被C^rΓMischenko丛扭曲的签名算子的解析指标类的存在性，并证明了K-同调签名类通过组装映射到签名指标类.在本文中，我们继续我们的研究，证明了签名指标类是不变的理性维特边界和分层同伦。我们也能够确定这个分析类的拓扑类似物的Mischenko对称签名最近定义的Banagl。最后，我们定义了Witt-Novikov高阶签名，并证明了我们的分析结果蕴含一个纯拓扑定理，即如果K-理论中的集合映射是有理内射的，则Witt-Novikov高阶签名是分层的同伦不变量.
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