Knotted families from graspers

TL;DR

This paper constructs explicit homotopy classes in embedding spaces of arcs and circles in manifolds of dimension four or higher, using string link families and analyzing their detection via the Goodwillie-Weiss Taylor tower.

Contribution

It provides explicit classes in homotopy groups of embedding spaces for manifolds of dimension at least four, detected through the Taylor tower, advancing understanding of embedding space topology.

Findings

Explicit classes in homotopy groups of embedding spaces constructed.

Detection of classes via the Taylor tower of Goodwillie and Weiss.

Families of string links in the $d$-ball used to produce these classes.

Abstract

For any smooth manifold $M$ of dimension $d\geq4$ we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $M$, in every degree that is a multiple of $d-3$, and show that they are detected in the Taylor tower of Goodwillie and Weiss. The classes are obtained from families of string links constructed in the $d$-ball.