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18
votes
Accepted
Loop spaces and infinite braids
Yes, $B \beta_\infty$ is homology equivalent to $\Omega^2_0 S^2$, the zero component of the double loop space of $S^2$. The map $B_\infty \to S_\infty$ induces the obvious stablisation map $\Omega^2_0 …
3
votes
Accepted
Group completion of a monoid (braid groups)
This is explained in my paper "Group-Completion", local coefficient systems, and perfection, Q. J. Math. (Quillen Memorial Issue) 64 (3) (2013) 795-803. A sufficient condition is for the monoid to be …