On torsion in the homology of the Torelli group

2025/10/29 by Vladimirov, Andrei
#20J06 (Secondary) #57M07 (Primary) 20J05 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.25728

Abstract

Let S_g be a closed, oriented surface of genus g, and let Mod(S_g) denote its mapping class group. The Torelli group I_g is the subgroup of Mod(S_g) consisting of mapping classes that act trivially on H₁(S_g). For any collection of pairwise disjoint, separating simple closed curves on S_g, the corresponding Dehn twists pairwise commute and determine a homology class in H_k(I_g), which is called an abelian cycle. We prove that the subgroup of H_k(I_g) generated by such abelian cycles is a ℤ/2ℤ-vector space for all k, and that it is finite-dimensional for k = 2 and g ≥ 4.

Citations

• The second rational homology of the Torelli group
• The second rational homology of the Torelli group is finitely generated
• The Dimension of the Torelli group
• The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group
• The rational cohomology ring of the moduli space of abelian 3-folds

Related

• The 2-torsion in the second homology of the genus 3 mapping class\n group
• Abelian covers of surfaces and the homology of the Torelli group
• On the degree-two part of the associated graded of the lower central series of the Torelli group
• Generating the Torelli group
• Torsion Elements in the Mapping Class Group of a Surface