, Volume 130, Issue 2, pp 257-275

On the homotopy of the stable mapping class group

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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.

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