Algebraic & Geometric Topology

The Whitehead group and the lower algebraic $K$–theory of braid groups on $\mathbb{S}^2$ and $\mathbb{R}P^2$

Daniel Juan-Pineda and Silvia Millan-López

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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.

Article information

Algebr. Geom. Topol., Volume 10, Number 4 (2010), 1887-1903.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19A31: $K_0$ of group rings and orders 19B28: $K_1$of group rings and orders [See also 57Q10]
Secondary: 55N25: Homology with local coefficients, equivariant cohomology

Whitehead group braid group


Juan-Pineda, Daniel; Millan-López, Silvia. 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 (2010), no. 4, 1887--1903. doi:10.2140/agt.2010.10.1887.

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