Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups

2025/09/03 by Holden, Annie · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2509.03742

Abstract

We introduce two Torelli subgroups of the handlebody group. The group HI_g,p^b is the subgroup of the handlebody group acting trivially on the first homology of the boundary surface, and H_B I_g,p^b is the subgroup of the handlebody group acting trivially on the first homology of the handlebody. Using the symplectic representation and the Johnson homomorphisms for the Torelli subgroups of the mapping class group and of Aut(F_g), we define abelian quotients of these handlebody Torelli groups. In terms of the representation theory of the special linear group, we describe cup products of two classes in the first rational cohomology groups of HI_g,p^b and H_B I_g,p^b obtained by the rational duals of these abelian quotients.

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• The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

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