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On the homotopy of the stable mapping class group

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Inventiones mathematicae Aims and scope


By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ +, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ + coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ + to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ +≃Im J (1/2)×Y, where Im J (1/2) is the image of J localized away from the prime 2.

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Oblatum 23-X-1995 &19-XI-1996

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Tillmann, U. On the homotopy of the stable mapping class group. Invent math 130, 257–275 (1997).

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