Abstract
AbstractIn this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts.
Highlights
AND PRELIMINARIESOur main results are the definition and computation of coarse homotopy and coarse homotopy groups, in the category of generalized coarse spaces, as introduced in particular by John Roe [8].In this note, we discuss in detail the concept of coarse homotopy
We introduce a geometric version of coarse homotopy groups and show their basic properties
The main computation is the calculation of the coarse homotopy groups of cones on simplicial complexes: they are equal to the homotopy groups of the base of the cone
Summary
Our main results are the definition and computation of coarse homotopy and coarse homotopy groups, in the category of generalized coarse spaces, as introduced in particular by John Roe [8]. We introduce a geometric version of coarse homotopy groups and show their basic properties (in particular that they form groups in the first place). The main computation is the calculation of the coarse homotopy groups of cones on simplicial complexes: they are equal to the homotopy groups of the base of the cone. Preliminary results in this direction are contained in the Göttingen doctoral thesis of Behnam Norouzizadeh [6]. Before we get to these results, we start with preliminaries, introducing the coarse category and deriving some gluing theorems for coarse maps, which are indispensable when working with geometric homotopy groups.
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