Descent and the $KH$-assembly map
Paul D. Mitchener
在本文中，我们证明了粗糙几何中下降的一般概念可以应用于$KH$ -集合映射的内射性研究。我们还表明，粗组装映射是内射一般有限粗$CW$ -复合物。
In this article we show that a general notion of descent in coarse geometry can be applied to the study of injectivity of the $KH$-assembly map. We also show that the coarse assembly map is injective in general for finite coarse $CW$-complexes.
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We show that for finite groups the Loday assembly map with coefficients in finite fields is in general not injective.
We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K- and L-theory of group rings and group C^*-algebras predict that certain assembly maps are weak homotopy equivalences.
The Baum-Connes assembly map and the generalized Bass conjecture
证明了Connes-Karoubi-Chern特征标的象，仅限于复群代数的Bott-periodized拓扑K理论中的Baum-Connes集结映射的象，存在于群代数的（周期）循环同调的椭圆和项中.这意味着对于群代数的任何（加权）ell-1完备化，Baum-Connes集合映射的有理满射性意味着该代数的广义Bass猜想。
The Behavior of Nil-Groups under Localization and the Relative Assembly Map
研究了K-群的诣零子群在局部化下的行为。由此得到有限子群族到虚循环子群族的相对组装映射是有理同构。结合等变的Chern特征标，利用群同调和有限循环子群的K-群给出了K-理论集结映射的有理化源的完整计算.
KK-theory of C*-categories and the analytic assembly map
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。
On rational injectivity of Kasparovs assembly map in dimension <=2
利用Kasparov的有理化装配映射到满群C^*-代数C^*（G）（用有理数张量）的（算子）K-理论中，给出了应用于维数小于或等于2的（可数）离散群G的分类空间BG的有理陈特征标的逆的合成在同调中的内射性的新证明.
Coarse and equivariant co-assembly maps
我们研究了一个等变共组装映射，它与通常的Baum-Connes组装映射是对偶的，与粗糙几何、等变Kasparov理论和对偶Dirac态射的存在性密切相关。作为应用，我们证明了对偶狄拉克态射的存在性与适当的紧化，即满足Carlsson-Pedersen条件的群体，我们研究了一个K-理论对应的适当Lipschitz上同调的Connes，Gromov和Moscovici。
Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-the…
Dualizing the coarse assembly map
我们制定和研究一个新的粗（共）组装映射。它涉及修改的希格森电晕建设，并产生一个地图双在适当的意义上的标准粗组装地图。新的装配图在许多情况下是同构的。对于一个群的底层度量空间，粗糙的共组装映射与对偶狄拉克态射的存在性密切相关，因此与攻击诺维科夫猜想的狄拉克对偶狄拉克方法密切相关。
The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and Lück with the category theoretical construction by Meyer and Nest. This extends the result of Hambleton and Pedersen to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer's and Nest's identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson.
Trace class operators, regulators, and assembly maps in K-theory
设G是群，Fin是它的有限子群族，E（G，Fin）是分类空间.设L^1是复数上无限维可分Hilbert空间中迹类算子代数。考虑同伦代数K理论中的有理装配映射H_p^G（E（G，Fin），KH（L^1））\otimes \Q \to KH_p（L^1[G]）\otimes \Q .有理KH-同构猜想预言上面的映射是一个同构;它由Yu的定理（见arXiv：1106.3796，arXiv：1202.4999）得出，它总是单射的。在本文中，我们证明了以下几点。定理：假设上面的映射是满射的。设n \equiv p+1 \mod 2.则：i）平凡族H_n^G（E（G，{1}），K（\Z））\to K_n（\Z [G]）的装配映射是有理内射的. ii）对任意数域F，集合映射H_n^G（E（G，Fin），K（F））\to K_n（F[G]）是有理内射的.我们注意到，K理论Novikov猜想断言上述定理的第i）部分对所有G成立，第ii）部分等价于数域上的K理论Farrell-Jones猜想的有理内射性部分。定理证明的思想是使用Karoubi的乘法K理论的代数，等变版本，我们在这篇文章中介绍。
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.
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.
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。
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.
Split Injectivity of the Baum-Connes Assembly Map
本文利用Carlsson和Pedersen提出的连续控制方法证明了Baum-Connes映射是满足一定几何条件的群的分裂注入。
Higher Whitehead torsion and the geometric assembly map
利用空间的代数K理论，构造了一个高阶Whitehead挠率映射，并证明了它满足经典Whitehead挠率的一般性质.这用于以纯粹同伦理论术语描述定义在稳定的结构空间上的“几何装配映射”。
A descent principle for the Dirac dual Dirac method
设G是一个无挠的离散群，其分类空间BG是有限维的.我们证明了G有对偶Dirac态射当且仅当某个粗（共）集映射是同构。因此，这样的G的对偶Dirac态射的存在性是G的度量，即粗糙不变量。我们得到了类似的结果与挠群。我们开发的框架也适用于研究Lipschitz和适当的Lipschitz上同调的Connes，Gromov和Moscovici。
Continuous Control and the Algebraic L-theory Assembly Map
本文证明了L-理论中有限子群族的装配映射是一类群的分裂注入。这类群，包括实际上的多环群，具有满足某些几何条件的泛空间。证明遵循Carlsson-Pedersen开发的方法，在无挠的群的情况下分裂装配映射。在这里，连续控制的技术和结果扩展到处理组扭转。
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…
Finite asymptotic dimension and the coarse assembly map
该研究针对具有弱转移的强粗同调理论与弱有限同伦渐近维数的有界粗空间，给出粗装配映射为幻影等价这一结论的简单论证。
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…
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。
We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coad…
The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map
本文展示单位ary共轭群oid作为 Baum--Connes 装配映射的通用中介，扩展了 K_1 指数框架到非 Type I 情况，并通过 Morita 等价将单位ary 共轭群oid与变换群oid联系起来。
We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we…
Split injectivity of A-theoretic assembly maps
本文通过构造A-理论同调理论，推导出A-理论组装映射的分裂注入性结果，并证明Waldhausen的K-理论定理适用于非连通版本。
The $KH$-Isomorphism Conjecture and Algebraic $KK$-theory
本文证明了Bartels和L{ü}ck定义的$KH$ -Ask映射可以用Cortinas和Thom的代数$KK$ -理论来描述. $KH$ -装配图的$KK$ -理论描述与Baum-Connes装配图的描述相似。在非常基本的情况下，用于证明鲍姆-康纳斯猜想的方法也适用于$KH$ -同构猜想。
Operator ideals and assembly maps in $K$-theory
设$\cB$是复可分Hilbert空间中的有界算子环。对于$p>0$，考虑Schatten理想$\cL^p$，它由奇异值序列是$p$ -可和的那些算子组成;放入$\cS=\bigcup_p\cL^p$。设$G$是群，$\vcyc$是虚循环子群族.余国梁证明了$K$ -理论装配映射\[ H_*^G(\cE(G,\vcyc),K(\cS))\to K_*(\cS[G]) \]是有理内射的.他的证明涉及建设的一定陈德铭字符量身定制的工作与系数$\cS$和使用的一些结果有关代数$K$ -理论的经营者理想和控制拓扑和粗糙的几何。本文给出了Yu的结果的一个不同的证明。我们的证明使用了通常的Chern特征标来证明循环同调。像余的，它依赖于结果代数$K$ -理论的经营者理想，但没有控制的拓扑结构或粗糙的几何技术。我们制定的同伦$K$理论的结果。证明了有理装配映射\[ H_*^G(\cE(G,\fin),KH(\cL^p))\otimes\Q\to KH_*(\cL^p[G])\otimes\Q \]是单射的.我们证明了后一个映射等价于余所考虑的装配映射，从而得到了余的结果作为推论。
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。
K-theory for the maximal Roe algebra of certain expanders
本文研究了由剩余有限群产生的扩张图族的粗Baum-Connes组装映射的极大形式。不同于通常的Roe代数，我们表明，这是密切相关的装配图（最大）鲍姆-康纳斯装配图的组，是一个同构的一类扩展。我们还介绍了一个定量的Baum-Connes组装映射，并讨论了它的连接K-理论（极大）Roe代数。
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 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。
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。
Coarse co-assembly as a ring homomorphism
粗糙空间的稳定希格森日冕的$K$理论具有正则环结构。这个环是爱默生和迈耶的粗糙共组装图的未简化版本的域。我们证明了目标也具有环结构，并且在给定的粗糙空间在粗糙意义上是可收缩的条件下，联合组装是一个环同态。
We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular fo…
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。
We formulate and prove a Bott periodicity theorem for an $\ell^p$-space ($1\leq p<\infty$). For a proper metric space $X$ with bounded geometry, we introduce a version of $K$-homology at infinity, denoted by $K_*^{\infty}(X)$, and the Roe algebra at infinity, denoted by $C^*_{\infty}(X)$. Then the coarse assembly map descents to a map from $\lim_{d\to\infty}K_*^{\infty}(P_d(X))$ to $K_*(C^*_{\infty}(X))$, called the coarse assembly map at infinity. We show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show…
Filtrations of simplicial functors and the Novikov Conjecture
本文证明了离散群G的极大C*-代数C*（G）的强Novikov猜想等价于拓扑K理论中的一个命题，对于这个命题，代数K理论中的相应命题总是成立的.我们还证明了对任意群G，C*（G）的拓扑K理论的全集映射的有理内射性是由限制集映射的有理内射性导出的.
We study the $K$-theory and Swan theory of the group ring $R[G]$, when $G$ is a finite group and $R$ is any ring or ring spectrum. In this setting, the well-known assembly map for $K(R[G])$ has a companion called the coassembly map. We prove that their composite is the equivariant norm of $K(R)$. This gives a splitting of both assembly and coassembly after $K(n)$-localization, a new map between Whitehead torsion and Tate cohomology, and a partial computation of $K$-theory of representations in the category of spectra.
We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $Γ$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to …
Higher index theory for certain expanders and Gromov monster groups I
在本文中，第一个系列的两个，我们继续研究高指数理论的扩张。证明了如果一个图序列是扩张图且其围长趋于无穷大，则对于相应的度量空间$X$，粗Baum-Connes装配映射是内射的，但不是满射的.具有这种围长属性的扩展器是构建所谓的“Gromov怪物”群的必要成分，这些群（粗略地）在其凯莱图中包含扩展器。我们利用这种关系证明了具有一定系数的Baum-Connes装配映射对这些群是内射的而不是满射的。利用本系列第二篇论文的结果，我们还证明了具有这些系数的极大Baum-Cones装配映射是同构的。
Injectivity results for coarse homology theories
本文通过等效粗同调理论和降维原理，研究了线性群及具有有限分解复杂度群的装配映射单射性问题。
We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…
Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map
本文研究紧量子群的射影表示理论，证明其由正则酉2-上闭链诱导，并借此定义形变交叉积，最终获得了置换无挠离散量子群的量子Baum-Connes拼装映射。
Algebraic K-theory of Geometric Groups
本文通过同伦理论技术证明几何有限群的K-理论组装映射等价，并展示了该结果在群同调和几何中的应用。
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