On the homotopy of the stable mapping class group. (English) Zbl 0891.55019

Tillmann, Ulrike

Invent. Math. 130, No. 2, 257-275 (1997).

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.

Reviewer: J.Huebschmann (Villeneuve d’Ascq)

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

Keywords:

mapping class group; infinite loop spaces; J-morphism; group completion; conformal field theory

Citations:

Zbl 0618.57005; Zbl 0673.55009