Double L-theory

TL;DR

This paper introduces refined algebraic structures called double Witt and L-groups that capture more detailed invariants in linking forms and knot theory, leading to new tools for studying doubly-slice knots.

Contribution

It develops the theory of double Witt and L-groups, providing a more comprehensive algebraic framework and applying it to high-dimensional knot invariants and doubly-slice knots.

Findings

Double Witt groups detect infinitely many more signatures than single Witt groups.

A new exact sequence relates double L-groups to classical L-theory.

Every Seifert matrix for a doubly-slice knot is hyperbolic.

Abstract

We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal of the underlying ring, our double Witt groups capture infinitely many more integral signatures of the linking form than the single Witt groups. The double L-groups are an algebraic theory of `double cobordism', refining L-theory analogously. We exhibit an exact sequence relating the double L-groups to classical projective L-theory via a double homology surgery obstruction group. The algebraic techniques are applied to high-dimensional knot theory to define new invariants for the study of doubly-slice knots. In particular we prove a homomorphism from the n-dimensional double concordance group to a double L-group, which factors the construction of the Blanchfield form. Some results of Stoltzfus in this area are reproved and we show that every Seifert matrix for a doubly-slice knot is hyperbolic.