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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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