About torsion in the cokernels of the Johnson homomorphisms

2023/04/25 by Faes, Quentin
#20F34 #20F38 #57K20 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2304.12662

Abstract

The so-called Johnson homomorphisms (τ_k)_{k ≥ 1} embed the graded space associated to the Johnson filtration of a surface with one boundary component into the Lie ring of positive symplectic derivations D(H). In this paper, we show the existence of torsion in the cokernels of the Johnson homomorphisms, for all even degrees, provided the genus is big enough. The Satoh trace Tr is one of the known obstructions to be in the image of τ. For each degree, we define a map on Ker(Tr) ∩ D(H) whose image is 2-torsion, and which vanishes on the image of τ. We also give a formula to compute these maps.

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