A small normal generating set for the handlebody subgroup of the Torelli group

Abstract

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps . As an application, we give a normal generating set for the handlebody subgroup of the level d mapping class group of an orientable surface.

The handlebody group and the images of the second Johnson homomorphism

“…We also know some facts about the first term IA WD A \ J 1 . A generating set was described by Omori in [20]. He gives the following theorem, where HBP stands for "homotopical bounding pair", and a genus-h HBP-map is the composition T c ı T 1 of two Dehn twists, where c and d are essential simple closed curves cobounding a surface of genus h, cobounding an annulus in the handlebody, and not bounding disks in the handlebody.…”

“…Additionally, the Johnson filtration of M is separating, and so is its intersection with A; hence the study of the filtration .A \ J k / k 1 , including the determination of its associated graded algebra L k 1 .A \ J k /=.A \ J kC1 /, is also relevant for the study of the group A itself. As the Torelli group I (the subgroup of M acting trivially at the homological level) is the first term J 1 of the Johnson filtration, the question addressed here is the next natural step after the study of A \ I pursued by Omori in [20], and the earlier computation of .A \ J 1 /=.A \ J 2 / given by Morita in [17].…”

The handlebody group and the images of the second Johnson homomorphism

“…We also know some facts about the first term IA := A ∩ J 1 . A generating set was described by Omori in [15]. He gives the following theorem, where HBP stands for "homotopical bounding pair", and a genus-h HBP-map is the composition T c • T −1 d of two Dehn twists where c and d are essential simple closed curves cobounding a surface of genus h, cobounding an annulus in the handlebody, and not bounding disks in the handlebody.…”

“…Besides, the Johnson filtration of M is separating, and so is its intersection with A: hence the study of the filtration (A∩J k ) k≥1 , including the determination of its associated graded k≥1 A∩J k A∩J k+1 , is also relevant for the study of the group A itself. As the Torelli group I (the subgroup of M acting trivially at the homological level) is the first term J 1 of the Johnson filtration, the question adressed here is the next natural step after the study of A ∩ I pursued by Omori in [15], and the earlier computation of A∩J1 A∩J2 given by Morita in [12]. To get a more precise grasp of the intersection A ∩ J 2 , we use the Johnson homomorphisms (τ k ) k≥1 introduced in [4], trace-like operators, and the Casson invariant.…”

Generalized Johnson homomorphisms for extended N-series

“…Another natural filtration of the handlebody group H is the restriction (M k ∩H) k≥0 of the usual Johnson filtration (M k ) k≥1 of the mapping class group M, which was reviewed in §1.1. For instance, the first term H ∩ M 1 is the intersection of the handlebody group with the Torelli group, see [Pi09,Om19] for a generating system. This approach, which is evoked in [Hen18, §7] and considered e.g.…”