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ABELIAN CYCLES IN THE HOMOLOGY OF THE TORELLI GROUP

Part of: Fiber spaces and bundles Low-dimensional topology

Published online by Cambridge University Press:  18 October 2021

Erik Lindell Open the ORCID record for Erik Lindell [Opens in a new window]
Erik Lindell*
Affiliation:
Stockholm University, Roslagsvägen 101, Kräftriket 5-6, Matematik, 106 91, Stockholm, Sweden.
*
lindell@math.su.se
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Abstract

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism.

We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

Keywords

Torelli groupJohnson homomorphism

MSC classification

Primary: 55R40: Homology of classifying spaces, characteristic classes
Secondary: 57M99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 4 , July 2023 , pp. 1703 - 1726
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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