Arthur C. Bartels
We compare the domain of the assembly map in algebraic K-theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later.
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In this work, the assembly map in L-theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion.
We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.
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.
In this work, the continuously controlled techniques developed by Carlsson and Pedersen are used to prove that the Baum-Connes map is a split injection for groups satisfying certain geometric conditions.
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。
Generalizations of Loday's assembly maps for Lawvere's algebraic theories
本文将 Loday 装配映射推广到 Lawvere 代数理论，证明其 K-理论具有 lax 幺半结构，从而建立非阿贝尔插值方案并拓展应用范围。
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。
In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum-Connes conjecture. In its general form it may be constructed using Kasparov's equivariant KK-theory. In the special case of a torsionfree group the domain simplifies to the usual K-homology of the classifying space BG of G and it is frequently used that in this case the analytical assembly map is given by assigning to an operator an equivariant index. We give a precise formulation of this statement and prove it.
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…
We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficie…
In this work, the continuously controlled assembly map in algebraic $K$-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups $Γ$ that satisfy certain geometric conditions. The group $Γ$ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, $K_0(kΓ)$ is proved to be isomorphic to the colimit of $K_0(kH)$ over the finite subgroups $H$ of $Γ$, when $Γ$ is a virtually polycyclic group and $k$ is a field of characteristic zero.
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。
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…
We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through the assembly map of $A$-theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended $A$-theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type …
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。
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically exicisive functor to the coarsely excisive functor. We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along wit…
The algebraic K-theory of $k[\operatorname{SL}_2(\mathbb{F}_q)]$
利用迹方法计算了特征p完美域k上群环k[SL_2(F_q)]的高阶代数K-理论，核心是通过Lück-Reich-Rognes-Varisco的循环装配定理确定Sylow p-子群群环k[C_p^r]的代数K-理论，并发展了计算群环拓扑循环同调的工具。
A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach i…
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。
We show that the Strong Novikov Conjecture for the maximal C*-algebra C*(G) of a discrete group G is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group G, rational injectivity of the full assembly map for the topological K-theory of C*(G) follows from rational injectivity of the restricted assembly map.
We prove that for every finitely generated subgroup of a virtually connected Lie group which admits a finite dimensional model for the classifying space for proper actions the assembly map in algebraic K-theory is split injective. We also prove a similar statement for algebraic L-theory, which in particular implies the integral Novikov conjecture for such groups.
We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to \emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.
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 …
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