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2010 The Whitehead group and the lower algebraic $K$–theory of braid groups on $\mathbb{S}^2$ and $\mathbb{R}P^2$
Daniel Juan-Pineda, Silvia Millan-López
Algebr. Geom. Topol. 10(4): 1887-1903 (2010). DOI: 10.2140/agt.2010.10.1887


Let M=S2 or P2. Let PBn(M) and Bn(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 Γ, for Γ=PBn(M). The main results are that for Γ=PBn(S2), the Whitehead group of Γ, K̃0(Γ) and Ki(Γ) vanish for i1 and n>0. For Γ=PBn(P2), the Whitehead group of Γ vanishes for all n>0, K̃0(Γ) vanishes for all n>0 except for the cases n=2,3 and Ki(Γ) vanishes for all i1.


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Daniel Juan-Pineda. Silvia Millan-López. "The Whitehead group and the lower algebraic $K$–theory of braid groups on $\mathbb{S}^2$ and $\mathbb{R}P^2$." Algebr. Geom. Topol. 10 (4) 1887 - 1903, 2010.


Received: 18 September 2008; Revised: 14 June 2010; Accepted: 13 August 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1219.19004
MathSciNet: MR2728479
Digital Object Identifier: 10.2140/agt.2010.10.1887

Primary: 19A31, 19B28
Secondary: 55N25

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 4 • 2010
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