Browsing by Author "Day, Matthew"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item A Birman exact sequence for Aut(Fn)(Elsevier, 2012) Day, Matthew; Putman, AndrewThe 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.Item The complex of partial bases for Fn and nite generation of the Torelli subgroup of Aut(Fn)(Springer, 2013) Day, Matthew; Putman, AndrewWe study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of Aut(Fn) is nitely generated.