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

Juan-Pineda, Daniel; Millan-López, Silvia

doi:10.2140/agt.2010.10.1887

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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

DOI: 10.2140/agt.2010.10.1887

Abstract

Let $M = S^{2}$ or $ℝ P^{2}$ . Let $P B_{n} (M)$ 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 $ℤ Γ$ , for $Γ = P B_{n} (M)$ . The main results are that for $Γ = P B_{n} (S^{2})$ , the Whitehead group of $Γ$ , ${\tilde{K}}_{0} (ℤ Γ)$ and $K_{i} (ℤ Γ)$ vanish for $i \leq - 1$ and $n > 0$ . For $Γ = P B_{n} (ℝ P^{2})$ , the Whitehead group of $Γ$ vanishes for all $n > 0$ , ${\tilde{K}}_{0} (ℤ Γ)$ vanishes for all $n > 0$ except for the cases $n = 2, 3$ and $K_{i} (ℤ Γ)$ vanishes for all $i \leq - 1$ .

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

Volume 10, issue 4 (2010)