Controlled $K$-theory and $K$-homology

Motivated by the idea that our access to 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, known as the localization algebra, and 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 as done in the definition of the localization algebra, we prove that for each dimension n , there exists a threshold r_{n}>0 such that the K -homology of an 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 .