ON THE HOMOTOPY OF THE STABLE MAPPING CLASS GROUP

Authors
Citation
U. Tillmann, ON THE HOMOTOPY OF THE STABLE MAPPING CLASS GROUP, Inventiones Mathematicae, 130(2), 1997, pp. 257-275
Citations number
18
Categorie Soggetti
Mathematics, General",Mathematics
Journal title
ISSN journal
00209910
Volume
130
Issue
2
Year of publication
1997
Pages
257 - 275
Database
ISI
SICI code
0020-9910(1997)130:2<257:OTHOTS>2.0.ZU;2-#
Abstract
By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, B Gamma(infinity)(+), has the homotopy type of an infinite loop space. The main new tool is a generalized group co mpletion theorem for simplicial categories. The first deloop of B Gamm a(infinity)(+) coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping c lass group onto Siegel's modular group is shown to induce a map of inf inite loop spaces from B Gamma(infinity)(+) to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there i s a space Y such that B Gamma(infinity)(+) similar or equal to ImJ((1/ 2)) x Y, where ImJ((1/2)) is the image of J localized away from the pr ime 2.