Double L–groups and doubly slice knots

Orson, Patrick

doi:10.2140/agt.2017.17.273

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Double $L$–groups and doubly slice knots

Patrick Orson

Algebraic & Geometric Topology 17 (2017) 273–329

DOI: 10.2140/agt.2017.17.273

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 $n \geq 1$ .

Keywords

knot theory, L-theory, doubly slice, high-dimensional knot, Blanchfield pairing

Mathematical Subject Classification 2010

Primary: 57Q45

Secondary: 57R67, 57Q60, 57R65

References

Publication

Received: 1 December 2015

Revised: 11 April 2016

Accepted: 21 May 2016

Published: 26 January 2017

Authors

Patrick Orson
Department of Mathematics Durham University Lower Mountjoy, Stockton Road Durham DH1 3LE United Kingdom
http://www.maths.dur.ac.uk/~vkdx72/

Volume 17, issue 1 (2017)