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On the quotients of mapping class groups of surfaces by the Johnson subgroups

Part of: Homotopy theory

Published online by Cambridge University Press:  27 November 2019

TOMÁŠ ZEMAN Open the ORCID record for TOMÁŠ ZEMAN [Opens in a new window]
TOMÁŠ ZEMAN*
Affiliation:
Postal Address: Mathematical Institute, Woodstock Road, Oxford, OX2 6GG. (Previous e-mail: tomas.zeman@maths.ox.ac.uk) Current Address: Dept. of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden. e-mail: tomas.zeman@math.su.se
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Abstract

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

MSC classification

Primary: 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 55P47: Infinite loop spaces
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 355 - 377
Copyright
© Cambridge Philosophical Society 2019

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