On the relative $L$-theory and the relative signature of PL manifolds with boundary

TL;DR

This paper introduces a new group-theoretic description of the relative structure group of PL manifolds with boundary, establishing a surgery exact sequence and connecting the relative L-group to K-theory.

Contribution

It provides a novel description of the relative structure group and a surgery exact sequence in the PL category, linking relative L-theory to K-theory.

Findings

New group structure for the relative structure group

Surgery exact sequence in the PL category

Additive map from relative L-group to K-theory

Abstract

In this paper, we give a new description of the group structure of the relative structure group of PL manifolds with boundary, and obtain a surgery exact sequence in the category of groups. Then we focus on the relative $L$-group of PL manifolds with boundary, and map it to the $K$-theory additively.