On the homotopy of the stable mapping class group (Q1374775)

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 \textit{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 \textit{R. M. Charney} and \textit{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.