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The universal surface bundle over the Torelli space has no sections

Chen, Lei (2021) The universal surface bundle over the Torelli space has no sections. Mathematische Zeitschrift, 298 (3-4). pp. 917-934. ISSN 0025-5874. doi:10.1007/s00209-020-02625-2.

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For g > 3, we give two proofs of the fact that the Birman exact sequence for the Torelli group 1 → π₁(S_g) → I_(g,1) → I_g → 1 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 UI_(g,n) → ^(Tu′g,n)BI_(g,n)(resp. UPI_(g,n) → ^(Tug,n)BPI_(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 ≥ 0 up to homotopy.

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Additional Information:© 2020 Springer-Verlag. Received 11 October 2017; Accepted 09 September 2020; Published 27 October 2020. 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 I2 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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Issue or Number:3-4
Record Number:CaltechAUTHORS:20201119-094222365
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Official Citation:Chen, L. The universal surface bundle over the Torelli space has no sections. Math. Z. 298, 917–934 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106735
Deposited By: Tony Diaz
Deposited On:19 Nov 2020 18:56
Last Modified:16 Nov 2021 18:56

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