Homomorphism obstructions for satellite maps

TL;DR

This paper investigates the conditions under which satellite maps in knot theory induce homomorphisms on concordance groups, providing new obstructions and verifying Hedden's conjecture in most cases.

Contribution

It introduces the first obstructions using Casson-Gordon signatures and constructs examples showing satellite maps often do not induce homomorphisms, supporting Hedden's conjecture.

Findings

Casson-Gordon signatures obstruct slice patterns inducing homomorphisms

Examples of satellite maps acting like homomorphisms without inducing them

Verification of Hedden's conjecture in the smooth category for most satellite operators

Abstract

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in $S^3$. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides $\pm 1$. We then provide subtle examples of satellite maps which map arbitrarily deep into the $n$-solvable filtration of [COT03], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden's conjecture in the smooth category for all but one small crossing number satellite operator.