代数$K$ -理论解析组合映射的谱与因子分解

Higher Whitehead torsion and the geometric assembly map

We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.

虚阿贝尔群的K理论的超补充集结

Hyperelementary assembly for K-theory of virtually abelian groups

控制$K$ -理论是用来表明，代数$K$ -理论的虚拟阿贝尔群所描述的组装映射使用可能无限的超互补子群。Farrell-Jones和项（来自无限子群）由群的有理射影空间参数化，并且简化版本是挠。包括关于装配和通用空间的一般材料。

Loday 装配映射对 Lawvere 代数理论的推广

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。

Algebraic K-theory of group rings and the cyclotomic trace map

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…

Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

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…

几何群的代数K-理论

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。

Operator ideals and assembly maps in $K$-theory

Let $\cB$ be the ring of bounded operators in a complex, separable Hilbert space. For $p>0$ consider the Schatten ideal $\cL^p$ consisting of those operators whose sequence of singular values is $p$-summable; put $\cS=\bigcup_p\cL^p$. Let $G$ be a group and $\vcyc$ the family of virtually cyclic subgroups. Guoliang Yu proved that the $K$-theory assembly map \[ H_*^G(\cE(G,\vcyc),K(\cS))\to K_*(\cS[G]) \] is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients $\cS$ and the use of some results about algebraic $K$-theory of ope…

等变粗同调理论与粗代数K-上同调

Equivariant coarse homotopy theory and coarse algebraic $\boldsymbol{K}$-homology

本文提出等变粗同调理论的公理框架，并引入等变Bornological粗空间和universal理论，探讨了等变粗代数K-上同调及其与锥 functor、组装映射的关系，为后续研究split-injectivity结果奠定基础。

$k[\operatorname{SL}_2(\mathbb{F}_q)]$ 的代数 $K$-理论

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。

相对装配映射与素特征下 Hecke 代数的 K 理论

Relative assembly maps and the K-theory of Hecke algebras in prime characteristic

本文研究了正则环系数扭群环代数K理论中从有限子群族到virtually cyclic子群族的相对装配映射，证明只需对特定素数取逆即可得到同构，并利用Witt向量模结构分析了素特征下约化p-进群Hecke代数的K理论。

Filtrations of simplicial functors and the Novikov Conjecture

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.

The algebraic and topological K-theory of the Hilbert Modular Group

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.

精化迁移、丛结构与代数K理论

The refined transfer, bundle structures and algebraic K-theory

我们给出了一个新的同伦理论准则，用以判定具有同伦有限纤维的纤维化何时可约化为具有紧拓扑流形纤维的纤维丛。这些准则导致了关于流形的同胚群的一个新的和意想不到的结果。证明中使用的工具是Waldhausen函子A（X）的集合映射的满射分裂。我们也给出了具体的例子，纤维化有一个减少到一个纤维丛与紧凑的拓扑流形纤维，但不承认一个紧凑的纤维光滑。实例由代数K理论不变量检测。我们考虑改进的贝克尔-戈特利布转移。我们表明，贝克尔和舒尔茨所描述的公理的一个版本，唯一确定的纤维化承认减少到一个纤维丛与紧凑的拓扑流形纤维类的精细转移。在附录中，我们概述了纤维化的特征类理论。这些类是找到紧凑的纤维平滑的主要障碍。