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Double $L$–groups and doubly slice knots

Patrick Orson

Algebraic & Geometric Topology 17 (2017) 273–329

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 1.

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knot theory, L-theory, doubly slice, high-dimensional knot, Blanchfield pairing
Mathematical Subject Classification 2010
Primary: 57Q45
Secondary: 57R67, 57Q60, 57R65
Received: 1 December 2015
Revised: 11 April 2016
Accepted: 21 May 2016
Published: 26 January 2017
Patrick Orson
Department of Mathematics
Durham University
Lower Mountjoy, Stockton Road
United Kingdom