Double $L$-groups and doubly-slice knots

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 $n\geq 1$.