Contents

1 Introduction

2 Fox Calculus

2.1 Fox Derivatives

2.2 The Magnus Representation for a Free Group

2.3 Homology and Cohomology of Groups

2.4 Fox Derivatives in Low-Dimensional Topology

3 Magnus Representations for Automorphism Groups of Free Groups

4 Magnus Representations for Mapping Class Groups

4.1 Definition

4.2 Symplecticity and Its Topological Interpretation

4.3 Non-faithfulness and Decompositions

4.4 Determinant of the Magnus Representation

4.5 The Johnson Filtration and Magnus Representations

4.6 Applications to Three-Dimensional Topology

5 Homology Cylinders

5.1 Definition and Examples

5.2 Stallings’ Theorem and the Johnson Filtration

6 Magnus Representations for Homology Cylinders I

6.1 Observation on Fundamental Groups of Homology Cylinders

6.2 The Acyclic Closure of a Group

6.3 Dehn-Nielsen Type Theorem

6.4 Extension of the Magnus Representation

7 Magnus Representations for Homology Cylinders II

8 Applications of Magnus Representations to Homology Cylinders

8.1 Higher-Order Alexander Invariants and Homologically Fibered Knots

8.2 Bordism Invariants and Signature Invariants

8.3 Abelian Quotients of Groups of Homology Cylinders

8.4 Generalization to Higher-Dimensional Cases