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Crossed modules

Johannes Huebschmann

Abstract

This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module. Crossed modules arise from the identities among the relations of the presentation of a group, from the extension problem for groups and, more generally, in low dimensional topology. Also, the (successful) attempt to extend the idea of a normal extension of commutative fields to the realm of non-commutative algebras leads to crossed modules. Crossed modules appear implicitly in a forgotten paper by A. Turing which in principle settles the extension problem for groups. Crossed modules make perfect sense for Lie algebras.

Crossed modules

Abstract

This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module. Crossed modules arise from the identities among the relations of the presentation of a group, from the extension problem for groups and, more generally, in low dimensional topology. Also, the (successful) attempt to extend the idea of a normal extension of commutative fields to the realm of non-commutative algebras leads to crossed modules. Crossed modules appear implicitly in a forgotten paper by A. Turing which in principle settles the extension problem for groups. Crossed modules make perfect sense for Lie algebras.
Paper Structure (11 sections, 23 equations)

This paper contains 11 sections, 23 equations.

Table of Contents

  1. Introduction
  2. Definition and basic examples
  3. Identities among relations
  4. Group extensions and abstract kernels
  5. Combinatorial group theory and low dimensional topology
  6. Interpretation of the third group cohomology group
  7. Higher group cohomology groups
  8. Generalization
  9. Lie algebra crossed modules
  10. Normality of a non-commutative algebra over its center
  11. Outlook

Theorems & Definitions (1)

  • proof