2014-10-01

# Discrete homology theory for metric spaces

## Publication

### Publication

*
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.

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Oxford U.P. | |

Bulletin of the London Mathematical Society | |

Combining Machine Learning and Game-theoretic Approaches for Cluster Analysis | |

Organisation | Networks and Optimization |

Barcelo, H., Capraro, V., & A. White, J. (2014). Discrete homology theory for metric spaces. Bulletin of the London Mathematical Society, 46, 889–905. |