Abstract.
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). https://doi.org/10.1007/s002220050184
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DOI: https://doi.org/10.1007/s002220050184