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.
Infinite loop spaces, and coherence for symmetric monoidal bicategories
