Abstract. Large-scale geometry, or coarse geometry, is the study of the asymptotic behaviour of a space, in contrast to the local properties studied in classical topology. Popularised by Gromov in his work in geometric group theory [Gro93], these ideas play a central role in modern geometric group theory, but also have an expanding reach to the broader context of general metric spaces [NY12]. In this thesis, we adopt the axiomatic framework of Roe’s coarse spaces [Roe03] as well as Mitchener, Norouzizadeh, and Schick’s notion of coarse homotopy [MNS20], where the authors define an asymptotic analogue of classical homotopy.
The central aim of this thesis is to further investigate this notion of coarse homotopy through invariants. We examine several existing coarse invariants, that is, properties invariant under a notion of large-scale equivalence, such as ends [Bri93] and asymptotic dimension [BD08], and determine whether they remain invariant under coarse homotopy. Particular attention is given to ends, where we draw comparisons to the notion of coarse path components introduced in [MNS20], as both ideas aim to classify the same asymptotic property, namely ‘connected components at infinity’.
Further, we introduce new coarse homotopy invariants by constructing coarse analogues of the classical fundamental groupoid and homotopy bigroupoid [HKK01]. In order to do this, we develop results on the interactions between metric cones and coarse cylinders enabling example computations. Finally, we conclude the penultimate and final chapters with results extending the main theorem of [MNS20], establishing a connection between the coarse fundamental groupoid and homotopy bigroupoid and their respective topological counterparts via metric cones.
