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I have checked everything "homology of loop spaces"-like, but was not able to find what is $H_*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?

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    $\begingroup$ You know it mod p for all primes, and it's not hard to figure out the Bockstein. Then there's a theorem of Cohen that the only p-torsion you'll see will have exact order p, so that should be enough. If I have time later I'll try to write out what happens. $\endgroup$ – Dylan Wilson Sep 9 '17 at 23:02
  • $\begingroup$ I guess Cohen says that the $E_{\infty}$-page (which as I said before is also the $E_2$-page) of the Bockstein sseq will look like $\mathbb{F}_p[y]$ where $|y|=1$? (yeah it's weird that we have a polynomial algebra without graded commutativity, and I'm not quite sure I'm parsing his result properly.) $\endgroup$ – Dylan Wilson Sep 9 '17 at 23:34
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There are homology isomorphisms $K(Br,1)\to \Omega^2_0 S^2$ and $\Omega^2 S^3\to \Omega^2_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br_n\hookrightarrow Br_{n+1}$).

As expected there is no neat description with integral coefficients, but much is known. You'll find a nice summary in Section 4 of this paper of Vershinin .

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The calculation you want is described in detail in Joe Neisendorfer's book Algebraic Methods in Unstable Homotopy Theory, Cambridge University Press, 2010. In particular, the Eilenberg--Moore spectral sequence collapses to give that $H_*(\Omega^2 S^3;\mathbb Z) = Cotor_*^{\mathbb Z[y]}(\mathbb Z, \mathbb Z)$. By calculating mod p, and then using the very simple Bockstein spectral sequence, he shows in Corollary 10.26.5, that $p$ annihilates the $p$--torsion for all primes $p$. Thus $H_*(\Omega^2 S^3;\mathbb Z)_{(p)}$ is a graded $\mathbb Z/p$ vector space (above dimension 1) whose Poincare series could easily be worked out from $H_*(\Omega^2 S^3;\mathbb Z/p)$.

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