Day, Matthew and Putman, Andrew. "A Birman exact sequence for Aut(Fn)." Advances in Mathematics, 231, (2012) Elsevier: https://hdl.handle.net/1911/71894.
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.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier.
A Birman exact sequence for Aut(Fn)
National Science Foundation
Advances in Mathematics
DMS-1005318 (National Science Foundation)