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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

Algebraic & Geometric Topology 10 (2010) 1887–1903
Abstract

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 i 1 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 i 1.

Keywords
Whitehead group, braid group
Mathematical Subject Classification 2000
Primary: 19A31, 19B28
Secondary: 55N25
References
Publication
Received: 18 September 2008
Revised: 14 June 2010
Accepted: 13 August 2010
Published: 17 September 2010
Authors
Daniel Juan-Pineda
Instituto de Matemáticas
Universidad Nacional Autónoma de México, Campus Morelia
Apartado Postal 61-3 (Xangari)
CP 58089
Morelia, Michoacán
Mexico
Silvia Millan-López
Facultad de Matemáticas
Campus Acapulco, Universidad Autónoma de Guerrero
Carlos E Adame No 54
Col La Garita
CP 39650
Acapulco, Guerrero
Mexico