图表空间的外部 Spanier-Whitehead 对偶性与同调表示定理

The third homology of stem-extensions and Whitehead's quadratic functor

Let $A \rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let $ρ: A\times G\to G$ be the multiplication map. We show that there is a natural map $φ: H_1(Σ_2^ε, {\rm Tor}_1^{\mathbb{Z}}({}_{2^\infty}A,{}_{2^\infty}A))\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$ such that, the image of $φ$ coincides with the image of the natural map $H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$. An important tool used here is Whitehead's quadratic functor $Γ$. As part of our proof of the main result, we give a precise homological …

K-Theoretic Duality for Hyperbolic Dynamical Systems

The K-theoretic analog of Spanier-Whitehead duality for noncommutative C*-algebras is shown to hold for the Ruelle algebras associated to irreducible Smale spaces. This had previously been proved only for shifts of finite type. Implications of this result as well as relations to the Baum-Connes conjecture and other topics are also considered.

$Local^{3}$ Index Theorem

$Local^{3}$ Index Theorem means $Local(Local(Local \;Index \; Theorem)))$. $Local \; Index \; Theorem$ is the Connes-Moscovici local index theorem \cite{Connes-Moscovici1}, \cite{Connes-Moscovici2}. The second "Local" refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. The Connes-Moscovici work is based on the operator $R(A) = \mathbf{P} - \mathbf{e}$ associated to the elliptic pseudo-differential operator $A$ on the smooth manifold $M$, where $\mathbf{P}$, $\mathbf{e}$ are idempotent…

Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces

Local Hochschild, cyclic Homology and K-theory were introduced by N. Teleman in [10] with the purpose of unifying different settings of the index theorem. This paper is one of the research topics announced in [10], §10. The definition of these new objects inserts the Alexander-Spanier idea for defining the co-homology [8] into the corresponding constructions. This is done by allowing only chains which have smal l support about the diagonal. This definition, applicable at least in the case of the Banach sub-algebras of the algebra of bounded operators on the Hilbert space of L2-sections in vect…

关于空间的周期同伦和同调等价

On periodic homotopy and homology equivalences of spaces

研究空间在染色高度n处的同伦理论，比较T(n)-等价与v_n-周期同伦群的概念，证明精确比较结果，并在无限循环空间中应用Kuhn的结果。

Smale空间KK-对偶中基本类的几何表示元

A geometric representative for the fundamental class in KK-duality of Smale spaces

该研究针对非交换几何中Smale空间KK-对偶的基本类表示问题，通过推广Bowen马尔可夫划分构造受控利普希茨常数的动力单位分解，结合非周期点采样得到非交换惠特尼嵌入定理类似物，最终构造出表示基本类的θ-可和Fredholm模。

图上自相似群胚作用的KK-对偶

KK-duality for self-similar groupoid actions on graphs

本文将Nekrashevych的KK-对偶从自相似群作用推广到图上自相似群胚作用，去除递归条件并将有限字母表推广为有限图，证明相关C*-代数与Smale空间的Ruelle代数强Morita等价，从而得到Spanier-Whitehead对偶。