In 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 cone of 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.

中文翻译：

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粗同伦群

在这篇关于粗几何的笔记中，我们重新审视了粗同伦。我们证明了粗同伦确实是一个等价关系，这在抽象粗结构的最一般情况下。我们引入（以几何方式）粗同伦群。主要结果是紧致单纯复形的锥粗同伦群与下层紧致单纯复形的通常同伦群重合。为了证明这一点，我们为锥体开发了几何三角测量技术，我们希望这些技术在不同的环境中也具有相关性。