Image of the third Johnson homomorphism

Quentin Faes; Ricard Riba

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arXiv · 2608.21956

Image of the third Johnson homomorphism

Quentin Faes·Ricard Riba

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Abstract

In this note we show that the image of the third Johnson homomorphism $\tau_3$ coincides with the kernel of Morita's trace map for the case of a surface of genus $g$ with one boundary component $\Sigma_{g,1}$, when $g \geq 6$. Moreover, as a consequence, we get that the image of $\tau_3$ on the handlebody subgroups coincides with the intersection of kernels of the Morita trace map and the Lagrangian trace maps.

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Quentin Faes, Ricard Riba. 2026-08-22. Image of the third Johnson homomorphism. https://arxiv.org/abs/2608.21956

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