Double L-groups and doubly-slice knots

الأصل · EN

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré 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≥ 1.