Algebraic & Geometric Topology 10 (2010) 1887–1903

DOI: 10.2140/agt.2010.10.1887

Abstract

Let M = S² or RP². Let PB_n(M) and B_n(M) be the pure and the full braid groups of M respectively. If Γ is any of these groups, we show that Γ satisfies the Farrell–Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic K–theory of the integral group ring ZΓ, for Γ = PB_n(M). The main results are that for Γ = PB_n(S²), the Whitehead group of Γ, K₀(ZΓ) and K_i(ZΓ) vanish for i ≤−1 and n > 0. For Γ = PB_n(RP²), the Whitehead group of Γ vanishes for all n > 0, K₀(ZΓ) vanishes for all n > 0 except for the cases n = 2,3 and K_i(ZΓ) vanishes for all i ≤−1.

Keywords

Whitehead group, braid group

Mathematical Subject Classification

Primary: 19A31, 19B28

Secondary: 55N25

References

Publication

Received: 18 September 2008
Revised: 14 June 2010
Accepted: 13 August 2010
Published: 17 September 2010

Authors