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On the homotopy of the stable mapping class group. (English) Zbl 0891.55019
The plus construction $$B\Gamma^+_{\infty}$$ of the classifying space of the stable mapping class group is shown to have the homotopy type of an infinite loop space. The basic tool is a folklore group completion theorem for simplicial categories (with constant object set) which does not seem to be in the literature and a proof of which is given in the appendix. The first delooping amounts to that of E. Y. Miller [J. Differ. Geom. 24, 1-14 (1986; Zbl 0618.57005)]. The well-known homomorphisms of the mapping class groups onto the modular groups are shown to induce a map of infinite loop spaces from $$B\Gamma^+_{\infty}$$ to the space $$B\roman{GL}(A)^+$$ for any topological algebra $$A$$ with unit. A result of R. M. Charney and F. R. Cohen [Mich. Math. J. 35, No. 2, 269-284 (1988; Zbl 0673.55009)] then entails a splitting of $$B\Gamma^+_{\infty}$$ as a product of the image of $$J$$ localized away from 2 with a certain complementary space.

MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 20F36 Braid groups; Artin groups 55P47 Infinite loop spaces 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 19D23 Symmetric monoidal categories 19L20 $$J$$-homomorphism, Adams operations 20J05 Homological methods in group theory 55Q50 $$J$$-morphism
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