Geometric Filtrations of Classical Link Concordance

TL;DR

This paper develops a filtration framework for classical link concordance using Whitney towers and gropes, revealing algebraic structures and obstructions related to framing and higher-order invariants.

Contribution

Introduces and analyzes grope and Whitney tower filtrations on link concordance, including twisted towers and their algebraic and obstruction-theoretic properties.

Findings

Associated graded quotients are finitely generated abelian groups

Higher-order Sato-Levine and Arf invariants obstruct framing

Relationships between twisted and framed filtrations are characterized by exact sequences

Abstract

This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) well-defined connected sum operation. Twisted Whitney towers are also introduced, along with a corresponding quadratic enhancement of the intersection theory for framed Whitney towers that measures Whitney-disk framing obstructions. The obstruction theory in the framed setting is strengthened, and the relationships between the twisted and framed filtrations are described in terms of exact sequences which show how higher-order Sato-Levine and higher-order Arf invariants are obstructions to framing a twisted Whitney tower. The results from this paper combine with those in \cite{CST2,CST3,CST4} to give a classifications of the filtrations; see our survey \cite{CST0} as well as the end of the introduction. UPDATE: This paper has been completely subsumed into the paper "Whitney tower concordance of classical links" \cite{WTCCL}.