ON THE K-THEORY OF SMALL MAPPING CLASS-GROUPS

The computations of cohomology for (classifying spaces of) some low ge nus mapping class groups in Benson and Cohen's recent American Mathema tical Society Memoir are applied, via the Atiyah-Hirzebruch spectral s equence, to find the K-theory. It is found that for the three groups i n question (4- and 6-punctured spheres and surface of genus 2) the red uced K-theory is profinite, being a direct sum of p-adic groups. There is a non-trivial K1 which cannot arise from representations; the part of K0 which arises via the symmetric group is identified, but in ever y case there is more, which raises the question of how other represent ations such as those defined by V. Jones might contribute.