#### Volume 10, issue 4 (2010)

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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 $M={\mathbb{S}}^{2}$ or $ℝ{P}^{2}$. 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 $ℤ\Gamma$, 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 $\Gamma$, ${\stackrel{̃}{K}}_{0}\left(ℤ\Gamma \right)$ and ${K}_{i}\left(ℤ\Gamma \right)$ vanish for $i\le -1$ and $n>0$. For $\Gamma =P{B}_{n}\left(ℝ{P}^{2}\right)$, the Whitehead group of $\Gamma$ vanishes for all $n>0$, ${\stackrel{̃}{K}}_{0}\left(ℤ\Gamma \right)$ vanishes for all $n>0$ except for the cases $n=2,3$ and ${K}_{i}\left(ℤ\Gamma \right)$ vanishes for all $i\le -1$.