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

TL;DR

This paper introduces a controlled $K$-theory approach to $K$-homology, providing a finite-resolution perspective that allows representing $K$-homology elements with finite matrices, bridging geometric and operator algebraic methods.

Contribution

It establishes a coarse graining version of Yu's localization algebra theorem using controlled $K$-theory, linking $K$-homology to operators with bounded propagation thresholds.

Findings

Proves $K$-homology is isomorphic to controlled $K$-theory groups with propagation bounds.

Shows $K$-homology elements can be represented by finite matrices.

Provides a finite-resolution framework for $K$-homology in simplicial complexes.

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 $X$.