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An infinite loop space structure on the nerve of spin bordism categories. (English) Zbl 1055.55011
For a connected compact oriented surface of genus \(g\) with \(n\) boundary circles \(F\), let \(\Gamma (F)=\Gamma_{g,n}\) denote the mapping class group of isotopy classes of orientation-preserving self-diffeomorphisms of \(F\) fixing the boundary pointwise, and a spin structure \({\mathfrak s}\) on \(F\) means a choice of a square root of the tangent complex line bundle of \(F\). For a spin surface \((F,{\mathfrak s})\), let \(G_{\mathfrak s}(F)\) denote the subgroup of \(\Gamma (F)\) consisting of all mapping classes \(f\in \Gamma (F)\) such that \(f^*{\mathfrak s}={\mathfrak s}\). Then Tillmann [U. Tillmann, Inv. Math. 130, No. 2, 257–275 (1997; Zbl 0891.55019)] proved that \(\mathbb Z \times \Gamma_{\infty}^{+}\) is an infinite loop space, where \(\Gamma_{\infty}\) denotes the direct limit of the group \(\Gamma_{g,1}\) formed by iterated attachment of a torus with two boundary components and \((-)^+\) Quillen’s plus-construction. In this paper, the author defines the stabilized spin mapping class group \(G_{\infty}\) in an analogous way by using Harer’s homology stabilization theorem [J. Harer, Math. Ann. 287, 323–334 (1990; Zbl 0715.57004)], and he proves that Tillmann’s stabilized theorem holds for the world of spin surfaces and spin mapping class groups. In particular, he shows that there is a homology equivalence \(\mathbb Z \times \mathbb Z/2\times BG_{\infty}\to \Omega ({\mathcal NS})\) and that the homology localization \(L_HBG_{\infty}\) of \(BG_{\infty}\) is an infinite loop space, where \({\mathcal NS}\) denotes the nerve of the spin bordism category \({\mathcal S}\). He also studies a variant of the spin mapping class group due to Masbaum [G. Masbaum, Geometry and Physiscs (Aarhus 1995), Lecture Notes in Pure and Applied Math. 184, Dekker, New York, 197–207 (1997; Zbl 0873.57011)], and he shows that its homology also stabilizes as the genus tends to infinity.

55P47 Infinite loop spaces
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57M99 General low-dimensional topology
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