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The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy theory, due to a result of Barratt-Priddy(-Segal(-Quillen(-others))) that "identifies" the sphere spectrum $QS^0$ with $S_{\infty}$.

In light of the short exact sequence above, are there any Barrat-Priddy-esque results on the infinite braid group $B_{\infty}?$ Is there even a loop space structure on $B_{\infty}?$

Ditto for the pure braids $P_{\infty}$, the kernal of $B_{\infty} \to S_{\infty}$.

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up vote 16 down vote accepted

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 $\Omega^2_0 S^2 \to Q_0S^0$.

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Do you know a proof (or where one is found)? – Romeo Oct 30 '10 at 20:49
I learnt it from G. Segal "Configuration-spaces and iterated loop-spaces" Invent. Math. 21 (1973) for example, special case b) after Theorem 3. I suspect this is not the earliest proof. – Oscar Randal-Williams Oct 30 '10 at 20:56
Ah, that's a great theorem! Do the pure braids fit in here? Fred Cohen has a result on the simplicial group $P_{\infty}$ containing a simplicial subgroup homotopy equivalent to $\Omega S^2$, but I'm not sure how/if that relates to your statement above. – Romeo Oct 30 '10 at 21:11
The homology of the pure braid groups does not stabilise in the usual sense, so the homology of $BP_\infty$ is going to be very large. However, you should look at the preprint "Representation theory and homological stability" by Church and Farb. – Oscar Randal-Williams Oct 30 '10 at 21:17
Closely related to this: there's a virtual vector bundle (classified by a map $B(B_\infty)\to BO$), such that the associated Thom spectrum is equivalent to the mod $2$ Eilenberg MacLane spectrum. That is, "mod 2 homolgy = braid bordism". See F. Cohen, "Braid orientations and bundles with flat connections", Invent. Math. 46 (1978). – Charles Rezk Oct 30 '10 at 22:40

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