#### Volume 17, issue 1 (2017)

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Double $L$–groups and doubly slice knots

### Patrick Orson

Algebraic & Geometric Topology 17 (2017) 273–329
##### Abstract

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 $L$–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 $L$–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 $n\ge 1$.

##### Keywords
knot theory, L-theory, doubly slice, high-dimensional knot, Blanchfield pairing
##### Mathematical Subject Classification 2010
Primary: 57Q45
Secondary: 57R67, 57Q60, 57R65