On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

2021/11/04 by Danica Kosanović, Kosanović, Danica
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2111.03041

Abstract

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M of dimension d≥4. In particular, if M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M, answering a question posed by Arone and Szymik. The case d=3 gives isotopy invariants of knots in a 3-manifold, that are universal of Vassiliev type ≤1, and reduce to Schneiderman's concordance invariant.

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