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

Abstract

Let $M={\mathbb{S}}^2$ or $ℝP^2$. Let $P{B}_n\left(M\right)$ and $B_n\left(M\right)$ be the pure and the full braid groups of $M$ respectively. If $\Gamma$ is any of these groups, we show that $\Gamma$ satisfies the Farrell–Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic $K$–theory of the integral group ring $\mathbb{Z}\Gamma$, for $\Gamma =P{B}_n\left(M\right)$. The main results are that for $\Gamma =P{B}_n\left({\mathbb{S}}^2\right)$, the Whitehead group of $\Gamma$, ${\tilde{K}}_0\left(\mathbb{Z}\Gamma \right)$ and $K_i\left(\mathbb{Z}\Gamma \right)$ vanish for $i\le -1$ and $n>0$. For $\Gamma =P{B}_n\left(ℝP^2\right)$, the Whitehead group of $\Gamma$ vanishes for all $n>0$, ${\tilde{K}}_0\left(\mathbb{Z}\Gamma \right)$ vanishes for all $n>0$ except for the cases $n=2,3$ and $K_i\left(\mathbb{Z}\Gamma \right)$ vanishes for all $i\le -1$.