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The universal surface bundle over the Torelli space has no sections

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Abstract

For \(g>3\), we give two proofs of the fact that the Birman exact sequence for the Torelli group

$$\begin{aligned} 1\rightarrow \pi _1(S_g)\rightarrow {\mathcal I}_{g,1}\rightarrow {\mathcal I}_g\rightarrow 1 \end{aligned}$$

does not split. This result was claimed by Mess (Unit tangent bundle subgroups of the mapping class groups, MSRI Pre-print, 1990), but his proof has a critical and unrepairable error which will be discussed in the introduction. Let \({\mathcal U\mathcal I}_{g,n}\xrightarrow {Tu'_{g,n}} {\mathcal B} {\mathcal I}_{g,n}\) (resp. \({\mathcal U}{\mathcal P}{\mathcal I}_{g,n}\xrightarrow {Tu_{g,n}}{\mathcal B}{\mathcal P}{\mathcal I}_{g,n}\)) denote the universal surface bundle over the Torelli space fixing n points as a set (resp. pointwise). We also deduce that \(Tu'_{g,n}\) has no sections when \(n>1\) and that \(Tu_{g,n}\) has precisely n distinct sections for \(n\ge 0\) up to homotopy.

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Acknowledgements

The author would like to thank Nick Salter and Jonathon Bowden for discussing the content of this paper. She thanks Matt Clay and Dan Margalit for reminding me of the fact that \({\mathcal I}_2\) is a free group. She would also like to extend her warmest thanks to Benson Farb for his extensive comments as well as for his invaluable support from start to finish. Lastly, she is also indebted to the anonymous referee for giving a complete and long list of corrections and suggestions, which makes the writing of the paper much better. The author is supported by NSF Grant DMS-2005409.

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  1. Department of Mathematics, Caltech, Pasadena, CA, USA

    Lei Chen

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  1. Lei Chen
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Chen, L. The universal surface bundle over the Torelli space has no sections. Math. Z. 298, 917–934 (2021). https://doi.org/10.1007/s00209-020-02625-2

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