A Birman exact sequence for Aut(F-n)

作者：Day Matthew; Putman Andrew^*

来源：Advances in Mathematics, 2012, 231(1): 243-275.

DOI：10.1016/j.aim.2012.04.024

• Free groups
• Automorphism groups of groups
• Automorphism groups of free groups
• Birman exact sequence
• Presentations of groups
• Recursive presentations of groups
• Cohomology of groups
• Johnson homomorphisms

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 keys to many of our proofs are several new generalizations of the Johnson homomorphisms.

• 出版日期2012-9-10