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Controlled $K$-theory and $K$-Homology

Ryo Toyota

Abstract

Motivated by the idea that our access to the spacetime is limited by the resolution of our measuring device, we give a new description of $K$-homology with a finite resolution. G. Yu introduced a $C^*$-algebra called the localization algebra $C^*_L(X)$ which consists of functions from $[1,\infty)$ to the Roe algebra $C^*(X)$ whose propagations converge to $0$ and he showed that for any finite dimensional simplicial complex $X$ endowed with the spherical metric, the $K$-theory of the localization algebra is isomorphic to the $K$-homology of $X$. We give a coarse graining version of this theorem using controlled $K$-theory (also known as quantitative $K$-theory). Namely, instead of considering families of operators whose propagations converge to $0$, we prove that for each dimension $n$, there exists a threshold $r_n>0$ such that the $K$-homology of $n$-dimensional finite simplicial complex $X$ is isomorphic to a certain group of equivalence classes of operators whose propagation is less than $r_n$. This picture also enables us to represent any element in the $K$-homology group $K_*(X)$ by a finite matrix for a finite simplicial complex $X$.

Controlled $K$-theory and $K$-Homology

Abstract

Motivated by the idea that our access to the spacetime is limited by the resolution of our measuring device, we give a new description of -homology with a finite resolution. G. Yu introduced a -algebra called the localization algebra which consists of functions from to the Roe algebra whose propagations converge to and he showed that for any finite dimensional simplicial complex endowed with the spherical metric, the -theory of the localization algebra is isomorphic to the -homology of . We give a coarse graining version of this theorem using controlled -theory (also known as quantitative -theory). Namely, instead of considering families of operators whose propagations converge to , we prove that for each dimension , there exists a threshold such that the -homology of -dimensional finite simplicial complex is isomorphic to a certain group of equivalence classes of operators whose propagation is less than . This picture also enables us to represent any element in the -homology group by a finite matrix for a finite simplicial complex .
Paper Structure (6 sections, 10 theorems, 101 equations)

This paper contains 6 sections, 10 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Roe algebras and controlled $K$-theory
  3. Overview of quantitative objects
  4. Functoriality and homotopy invariance of $K^{\varepsilon,r}_*(C^*(X))$
  5. Asymptotic Mayer-Vietoris exact sequence of controlled $K$-theory
  6. Quantitative description of $K$-homology

Key Result

Theorem 1.1

For each dimension $n$, there exist a constant $\lambda_n$, a function $h_n:(0,\frac{1}{4\lambda})\rightarrow [1,\infty)$, constants $r_n$ and $\varepsilon_n$ depending only on $n$ such that for any $n$-dimensional finite simplicial complex $X$, we have for all $(\varepsilon,r)$ with $0< \varepsilon< \varepsilon_n$ and $0<r<r_n$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Definition 2.6
  • Remark 2.2
  • Remark 2.3
  • ...and 35 more