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 the classical torsion algebraic –groups for localisations of a ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms.
We apply the double –groups in high-dimensional knot theory to define an invariant for doubly slice –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 –knots for .
"Double $L$–groups and doubly slice knots." Algebr. Geom. Topol. 17 (1) 273 - 329, 2017. https://doi.org/10.2140/agt.2017.17.273