Double L-theory

Abstract

This thesis is an investigation of the difference between metabolic and hyperbolic objects in a variety of settings and how they interact with cobordism and 'double cobordism', both in the setting of algebraic L-theory and in the context of knot theory. Let A be a commutative Noetherian ring with involution and S be a multiplicative subset. The Witt group of linking forms W(A,S) is defined by setting metabolic linking forms to be 0. This group is well-known for many localisations (A,S) and it is a classical fact that it forms part of a localisation exact sequence, essential to many Witt group calculations. However, much of the deeper 'signature' information of a linking form is invisible in the Witt group. The beginning of the thesis comprises the first general definition and careful investigation of the double Witt group of linking forms DW(A,S), given by the finer equivalence relation of setting hyperbolic linking forms to be 0. The treatment will include invariants, structure theorems and localisation exact sequences for various types of rings and localisations. We also make clear the relationship between the double Witt groups of linking forms over a Laurent polynomial ring and the double Witt group of those forms over the ground ring that are equipped with an automorphism. In particular we prove the isomorphism between the double Witt group of Blanchfield forms and the double Witt group of Seifert forms. In the main innovation of the thesis, we next define chain complex generalisations of the double Witt groups which we call the double L-groups DLn(A,S). In double L-theory, the underlying objects are the symmetric chain complexes of algebraic L-theory but the equivalence relation is now the finer relation of double algebraic cobordism. In the main technical result of the thesis we solve an outstanding problem in this area by deriving a double L-theory localisation exact sequence. This sequence relates the DL-groups of a localisation to both the free L-groups of A and a new group analogous to a 'double' algebraic homology surgery obstruction group of chain complexes over the localisation. We investigate the periodicity of the double L-groups via skew-suspension and surgery 'above and below the middle dimension'. We then reconcile the double L-groups with the double Witt groups, so that we also prove a double Witt group localisation exact sequence. Finally, in a topological application of double Witt and double L-groups, we apply our results to the study of doubly-slice knots. A doubly-slice knot is a knot that is the intersection of an unknotted sphere and a plane. We show that the double knot-cobordism group has a well-defined map to the DL-group of Blanchfield complexes and easily reprove some classical results in this area using our new methods.