Addendum to "Coarse homology theories"

Cyclic homology for bornological coarse spaces

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\mathbf{BornCoarse}$ to the cocomplete stable $\infty$-category of chain complexes $\mathbf{Ch}_\infty$. We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\mathcal{X} H^G$ by constructing a trace-like natural transformation $\mathcal{X} K_{k}^G\to…

打破等变粗同调理论的对称性

Breaking symmetries for equivariant coarse homology theories

本文提出粗几何中的对称性破缺构造，通过限制到更小群和空间获取等变粗同调类信息，并在等变粗K-同调中给出解析解释，应用于不变微分算子谱理论。

粗同调理论与有限分解复杂性

Coarse homology theories and finite decomposition complexity

本文通过粗同调理论探讨了具有等价有限分解复杂性的空间中forget-control映射纤维的消失性质。

半粗空间、同伦与同调

Semi-coarse Spaces, Homotopy and Homology

本文研究半粗空间的代数拓扑，构造同伦等价下不变的同调群，并证明无向图诱导的半粗同调与 Vietoris-Rips 同调同构，导出同伦不变性定理。

拓扑等变粗K-同调

Topological equivariant coarse K-homology

本文构造了具有严格群作用的$C^{*}$-范畴上的等变粗同调理论，通过引入受控于有界粗空间的Roe范畴并应用同调函子，验证了轨道范畴上的CP-函子性质，进而影响装配映射的单射性。

Equivariant Coarse (Co-)Homology Theories

We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all,…

The coarse Pimsner-Voiculescu sequence

We derive the Pimsner-Voiculescu sequence calculating the K-theory of a $C^{*}$- algebra with $\mathbb{Z}$-action using constructions with equivariant coarse K-homology theory. We then investigate to which extend this idea extends to more general equivariant coarse homology theories.

关于粗同调理论的单射结果

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。

粗同调论中的超越现象与 Chern 特征类

Transgressions and Chern characters in coarse homotopy theory

本文探讨了粗同调论中超越现象与 Chern 特征类之间的关系，研究了它们之间的可交换性及数学结构。

Secondary cup and cap products in coarse geometry

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $\mathrm{K}$-theory and -homology, the secondary products …

补空间的粗上同调

Coarse cohomology of the complement

该研究定义了度量空间子空间补集的粗（上）同调以推广Roe的理论，构建补集粗几何结构的模型空间，提出粗庞加莱对偶空间新研究方法，证明粗亚历山大对偶版本并给出粗PD($n$)空间的同调判别准则。

分支粗覆盖与转移映射

Branched coarse coverings and transfer maps

本文引入分支粗覆盖和转移映射概念，证明多种粗K-同调理论具有转移结构，并利用粗同伦理论中的Atiyah L²指标定理为Higson的粗Baum-Connes猜想反例提供新论证。

粗几何

Coarse geometry

本文综述粗几何，着重介绍粗同调理论这一核心方法，为该领域提供系统性概述。

粗粒化状态空间网络的持久同调

Persistent Homology of Coarse Grained State Space Networks

本文提出利用粗粒化状态空间网络结合持久同调进行动态状态检测，相比序数划分网络和时间延迟嵌入方法，在检测性能、噪声鲁棒性和计算效率上均有显著优势。

Coarse cohomology theories

We propose the notion of a coarse cohomology theory and study the examples of coarse ordinary cohomology, coarse stable cohomotopy and of coarse cohomology theories obtained by dualizing coarse homology theories. We show that the dualizing spectrum of a finitely generated torsion-free group only depends on the coarse motivic spectrum represented by the underlying bornological coarse space of the group. This in particular implies a conjecture of J. R. Klein that the dualizing spectrum of a group is a coarse invariant.

A-理论组装映射的分裂注入性

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。