The Whitehead group and the lower algebraicK–theory of braid groups on 𝕊2and ℝP2

Abstract

Let M D S or RP 2 . Let PBn.M / and Bn.M / be the pure and the full braid groups of M respectively. If € is any of these groups, we show that € satisfies the Farrell–Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic K–theory of the integral group ring Z€ , for € D PBn.M / . The main results are that for € D PBn.S/ , the Whitehead group of € , z K0.Z€/ and Ki.Z€/ vanish for i 1 and n > 0 . For € D PBn.RP / , the Whitehead group of € vanishes for all n > 0 , z K0.Z€/ vanishes for all n > 0 except for the cases nD 2; 3 and Ki.Z€/ vanishes for all i 1 .