Discrete homology theory for metric spaces
Bulletin of the London Mathematical Society , Volume 46 p. 889- 905
We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n n -dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg–Steenrod axioms, and prove a discrete analogue of the Mayer–Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a ‘fine enough’ rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.
|Keywords||homology theory, discrete metric spaces|
|THEME||Null option (theme 11)|
|Journal||Bulletin of the London Mathematical Society|
|Project||Combining Machine Learning and Game-theoretic Approaches for Cluster Analysis|
|Grant||This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/612.001.352 - Combining Machine Learning and Game-theoretic Approaches for Cluster Analysis|
Barcelo, H, Capraro, V, A. White, J, & Barcelo, H. (2014). Discrete homology theory for metric spaces. Bulletin of the London Mathematical Society, 46, 889–905.