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

Kosanović, Danica

doi:10.1090/tran/8805

On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant
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by Danica Kosanović;

Trans. Amer. Math. Soc. 377 (2024), 775-805

DOI: https://doi.org/10.1090/tran/8805

Published electronically: November 20, 2023

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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 $d\geq 4$. 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 $\leq 1$, and reduce to Schneiderman’s concordance invariant.

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