Mapping Graph Homology to $K$-Theory of Roe Algebras

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL Russian Journal of Mathematical Physics Pub Date : 2024-03-19 DOI:10.1134/S106192084010102

V. Manuilov

引用次数: 0

Abstract

Given a graph $\Gamma$, one may consider the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\Gamma$ and their $K$-theory counterparts — the $K$-theory of the (uniform) Roe algebra of the metric space $X$ of vertices of $\Gamma$. We construct here a natural mapping from homology of $\Gamma$ to the $K$-theory of the Roe algebra of $X$, and its uniform version. We show that, when $\Gamma$ is the Cayley graph of $\mathbb Z$, the constructed mappings are isomorphisms.

DOI 10.1134/S106192084010102

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将图同调映射到 $$K$$ - Roe 算法理论

摘要给定一个图（$\Gamma$），我们可以把它的顶点集（$X$）看作一个度量空间，假设所有的边的长度都是一。我们考虑两个版本的 $\Gamma$ 的同调理论和它们的 $K$ 理论对应物-- $\Gamma$ 的顶点的度量空间 $X$ 的（统一）Roe代数的 $K$ 理论。我们在这里构建了从$\Gamma$的同调到$X$的Roe代数的$K$-理论及其统一版本的自然映射。我们证明了当$\Gamma$ 是$\mathbb Z$ 的 Cayley 图时，所构造的映射是同构的。 doi 10.1134/s106192084010102

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.

Mapping Graph Homology to \(K\)-Theory of Roe Algebras

Abstract