Signatures in algebra, topology and dynamics

TL;DR

This paper surveys the historical development and modern applications of the signature of quadratic forms in algebra, topology, and dynamics, highlighting its significance across multiple mathematical disciplines.

Contribution

It provides a comprehensive overview of the evolution of the signature concept and its applications, including new insights and an appendix on algebraic L-theory.

Findings

Historical analysis of 19th-century signature development

Applications to topology of manifolds and dynamical systems

Inclusion of algebraic L-theory and localization sequences

Abstract

We survey the 19th century development of the signature of a quadratic form, and the applications in the 20th and 21st century to the topology of manifolds and dynamical systems. Version 2 is an expanded and corrected version of Version 1, including an Appendix by the second named author "Algebraic L-theory of rings with involution and the localization exact sequence".