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-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.

Barcelo, H., Capraro, V., White, J. (2014). Discrete homology theory for metric spaces. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 46(Part 5), 889-905 [10.1112/blms/bdu043].

Discrete homology theory for metric spaces

Capraro V;
2014

Abstract

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-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.
Articolo in rivista - Articolo scientifico
HOMOTOPY-THEORY
English
889
905
17
Barcelo, H., Capraro, V., White, J. (2014). Discrete homology theory for metric spaces. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 46(Part 5), 889-905 [10.1112/blms/bdu043].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/397911
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