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On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups

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Abstract

The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group.

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Acknowledgements

We would like to thank the referee for constructive comments and questions, and Lei Chen, Benson Farb, Jesús Hernández Hernández and Justin Lanier for useful email communications.

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Authors and Affiliations

  1. Mathematics Department, Northeastern University, Boston, MA, 02115, USA

    Solomon Jekel

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca de Juárez, Oaxaca, 68000, Mexico

    Rita Jiménez Rolland

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  1. Solomon Jekel
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  2. Rita Jiménez Rolland
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Correspondence to Rita Jiménez Rolland.

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Rita Jiménez Rolland is grateful for the financial support from PAPIIT DGAPA-UNAM grant IA104010 and from CONACYT grant CB-2017-2018-A1-S-30345-F-3125.

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Jekel, S., Rolland, R.J. On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups. Arnold Math J. 7, 159–168 (2021). https://doi.org/10.1007/s40598-020-00159-3

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  • DOI: https://doi.org/10.1007/s40598-020-00159-3

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