A Birman exact sequence for Aut(F_n)

TL;DR

This paper constructs an analogous exact sequence for Aut(F_n), showing the kernel is finitely generated but not finitely presentable, with infinite rank second rational homology, extending Johnson homomorphisms.

Contribution

It introduces a new exact sequence for Aut(F_n), analyzes the kernel's algebraic properties, and generalizes Johnson homomorphisms for free groups.

Findings

Kernel is finitely generated

Kernel has infinite rank second rational homology

Constructed an explicit infinite presentation

Abstract

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The key to many of our proofs are several new generalizations of the Johnson homomorphisms.