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

TL;DR

This paper computes the fundamental homotopy groups of embedding spaces of arcs and circles in high-dimensional manifolds, revealing connections to homology and knot invariants, and providing answers to existing mathematical questions.

Contribution

It explicitly calculates the first homotopy groups of embedding spaces in various cases, linking them to homology groups and knot invariants, and addresses open questions in the field.

Findings

Fundamental group of embedding spaces in simply connected 4-manifolds equals second homology.

In 3-dimensions, provides universal Vassiliev-type invariants for knots.

Connects the Dax invariant to Schneiderman's concordance invariant.

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 $d\geq4$. 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 $\leq1$, and reduce to Schneiderman's concordance invariant.