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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.

MSC:
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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