Clasper Concordance, Whitney towers and repeating Milnor invariants

TL;DR

This paper establishes a correspondence between Whitney towers and clasper surgeries in 4-dimensional topology, classifies link concordance filtrations using Milnor invariants, and introduces twisted Whitney towers for generalized link homotopy.

Contribution

It proves that links bounding degree k Whitney towers are exactly those related to the unlink via C_k-concordance and clasper surgeries, linking these concepts through tree structures.

Findings

Whitney towers correspond to clasper surgeries in the 4-ball.

C_k-concordance classifies links via Milnor and higher-order invariants.

Introduces k-repeating twisted Whitney towers for generalized link homotopy.

Abstract

We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants. Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted \emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants.