How do Segal's theorems from (Configuration-spaces and iterated loop-spaces. Invent. Math.21:213--221) imply that there is an isomorphism $H_*(B_\infty,\mathbb{Z})\cong H_*(\Omega^2S^3,\mathbb{Z})$, where $B_\infty=\varinjlim B_n$ and $B_n$ is the $n$--strings Braid group?
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Looping the fiber sequence $S^1 \to S^3 \to S^2$ gives $\Omega^2 S^2 \simeq \mathbb{Z} \times \Omega^2 S^3$. This is the group completion $\mathbb{Z} \times BB_\infty^+$ of $\coprod_{n\ge0} BB_n$, so by Segal's theorem $\bigoplus_{n \in \mathbb{Z}} H_*(B_\infty)$ is isomorphic to $H_*(\Omega^2 S^2) \cong \bigoplus_{n \in \mathbb{Z}} H_*(\Omega^2 S^3)$.
