Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1377–1394 | Cite as

Cup products in surface bundles, higher Johnson invariants, and MMM classes

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Abstract

In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group \({\text {Mod}}_g\) and the Torelli group \(\mathcal {I}_g\). We show that Kawazumi’s twisted MMM class \(m_{0,k}\) can be used to compute k-fold cup products in surface bundles, and that \(m_{0,k}\) provides an extension of the higher Johnson invariant \(\tau _{k-2}\) to \(H^{k-2}({\text {Mod}}_{g,*}, \wedge ^k H_1)\). These results are used to show that the behavior of the restriction of the even MMM classes \(e_{2i}\) to \(H^{4i}(\mathcal {I}_g^1)\) is completely determined by \({\text {im}}(\tau _{4i}) \le \wedge ^{4i+2}H_1\), and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel \(\mathcal K_{g,*}\) have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all \(\tau _i\) when restricted to \(\mathcal K_{g,*}\).

Notes

Acknowledgements

Many thanks are due to Madhav Nori, for the inspiring conversations that sparked my interest in and approach to this problem. I would also like to thank Ilya Grigoriev and Aaron Silberstein for helpful discussions along the way. As always, this paper would not have been possible without continued interest, support, and guidance from Benson Farb, as well as many comments on preliminary drafts.

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© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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