The Whitehead group and the lower algebraic K–theory of braid groups on 𝕊2 and ℝP2

Juan-Pineda, Daniel¹ ; Millan-López, Silvia²

¹Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3 (Xangari), CP 58089, Morelia, Michoacán, Mexico
²Facultad de Matemáticas, Campus Acapulco, Universidad Autónoma de Guerrero, Carlos E Adame No 54, Col La Garita, CP 39650, Acapulco, Guerrero, Mexico

Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 1887-1903

Résumé

Let M = S2 or ℝP2. 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 ℤΓ, for Γ = PBn(M). The main results are that for Γ = PBn(S2), the Whitehead group of Γ, K̃0(ℤΓ) and Ki(ℤΓ) vanish for i ≤−1 and n > 0. For Γ = PBn(ℝP2), the Whitehead group of Γ vanishes for all n > 0, K̃0(ℤΓ) vanishes for all n > 0 except for the cases n = 2,3 and Ki(ℤΓ) vanishes for all i ≤−1.

Détail

DOI : 10.2140/agt.2010.10.1887

Keywords: Whitehead group, braid group

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