Codimension 2 transfer of signatures in L theory

TL;DR

This paper establishes a transfer map in L-theory for signatures of manifolds, extending previous K-theoretic results to the L-theoretic setting with controlled torsion, under certain homotopy conditions.

Contribution

It constructs a transfer map between symmetric L-groups of fundamental groups, answering a question about codimension 2 invariance of signatures.

Findings

Constructed a transfer map in L-theory for signatures.

Proved the map carries the signature of M to that of N up to torsion.

Extended invariance results from K-theory to L-theory.

Abstract

The signature of a closed manifold is an important geometric topology. Let $M$ be a closed manifold and $N$ be a codimension 2 submanifold of it. Given certain homotopy conditions, Higson, Xie and Schick proved an invariance theorem in codimension 2 for the $K$-theoretic signature. They asked for the $L$-theoretic counterpart of their result. In this note, we will answer their question and moreover, construct a tranfer map between the symmetric $L$-groups of the fundamental groups of $M$ and $N$, which carries the signature of $M$ to that of $N$ up to a torsion of order at most $4$.