A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces

TL;DR

This survey reviews Magnus representations, focusing on their applications to mapping class groups and homology cobordisms of surfaces, highlighting foundational concepts and recent research applications.

Contribution

It provides a comprehensive overview of Magnus representations and their extensions to homology cobordisms, connecting classical theory with recent developments.

Findings

Magnus representations are useful in studying automorphism groups of free groups.

Applications include insights into mapping class groups and homology cobordisms.

Recent research has expanded the theory's scope and applications.

Abstract

This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups and mapping class groups of surfaces with related topics. In the latter half, we discuss in detail how the theory in the first half extends to homology cobordisms of surfaces and present a number of applications from recent researches.