Continuous Control and the Algebraic L-theory Assembly Map

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…

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。

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…

$k[\operatorname{SL}_2(\mathbb{F}_q)]$ 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。

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.

On the K-theory of subgroups of virtually connected Lie groups

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.

有理 U(2)-谱的一个代数模型

An algebraic model for rational U(2)-spectra

本文通过组装先前工作中的七个块（包括环面部分和小环面群的研究），结合融合群与 Weyl 群的详细输入，构造了一个明确且可计算的有理 U(2)-谱代数模型。

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.

Equivariant stable homotopy methods in the algebraic K-theory of infinite groups

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basic idea to the core and follow the evolution over time in order to explain the advantages of the flexible state that exists today. We end with an outline of the proof of the Borel conjecture in algebraic K-theory for groups of finite asymptotic dimension. We also discuss the relat…

到$\boldsymbol{\rm{P}}^m$的代数映射与连续映射族

Families of algebraic and continuous maps to $\mathbb{P}^m$

该研究阐述了如何利用比较到射影空间的代数映射与连续映射空间同调的结果，实现对代数映射族与连续映射族模叠的比较。

Splitting With Continuous Control in Algebraic K-theory

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.

Spaces of algebraic and continuous maps between real algebraic varieties

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration b…

Equivariant algebraic K-theory of G-rings

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…

Smoothing maps into algebraic sets and spaces of flat connections

Let X be a real algebraic subset of R^n and M a smooth, closed manifold. We show that all continuous maps from M to X are homotopic (in X) to C^\infty maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds.