On homotopy groups of spaces of embeddings of an arc or a circle: the Dax 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 $d\ge 4$. 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 $\le 1$, and reduce to Schneiderman's concordance invariant.

25 pages, 10 figures. Version accepted for publication in Trans. Amer. Math. Soc

References in corpus (3)

• A family of embedding spaces
• Knotted 3-balls in S^4
• Spaces of knotted circles and exotic smooth structures

Cited by in corpus (4)

• Knotted families from graspers
• Diffeomorphisms of 4-manifolds from graspers
• A new approach to light bulb tricks: Disks in 4-manifolds
• On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold