Double $L$-groups and doubly-slice knots

TL;DR

This paper introduces a refined algebraic framework called double L-groups, extending classical L-theory, and applies it to high-dimensional knot theory to define invariants for doubly-slice knots, establishing key properties of these invariants.

Contribution

It develops a new double-cobordism theory for chain complexes with Poincaré duality and applies it to define invariants for doubly-slice knots, refining existing algebraic invariants.

Findings

Double L-groups refine classical algebraic L-groups.

The new invariants detect doubly-slice properties of knots.

Stable doubly-slice implies doubly-slice for key algebraic forms.

Abstract

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincar\'{e} duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms. We apply the double $L$-groups in high-dimensional knot theory to define an invariant for doubly-slice $n$-knots. We prove that the "stably doubly-slice implies doubly-slice" property holds (algebraically) for Blanchfield forms, Seifert forms and for the Blanchfield complexes of $n$-knots for $n\geq 1$.