Calculating the image of the second Johnson-Morita representation

Joan S. Birman; Tara E. Brendle; Nathan Broaddus

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arXiv · 0708.3861

Calculating the image of the second Johnson-Morita representation

Joan S. Birman·Tara E. Brendle·Nathan Broaddus

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Abstract

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \wedge^3 H \semi \sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \wedge^3 H \semi \sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.

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BibTeX RIS

Joan S. Birman, Tara E. Brendle, Nathan Broaddus. 2007-08-28. Calculating the image of the second Johnson-Morita representation. https://arxiv.org/abs/0708.3861

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