Integer homology of double loop space of odd-dimensional sphere

Question

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?

Answer 1

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 .

Answer 2

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 $H_*(\Omega^2 S^3;\mathbb Z/p)$.

Answer 3

The next two claims completely describe $H_*(\Omega^2S^3;\mathbb{Z})$. This follows from several sources. For example, from already mentioned in the answer of Nicholas Kuhn book of Joe Neisendorfer.

Theorem 1. The space $H_*(\Omega^2S^3;\mathbb{Z}_p)$ is a primitively generated Hopf algebra such that $$ H_*(\Omega^2S^3;\mathbb{Z}_p)= \begin{cases} \Lambda_p[x_0,x_1,x_2,\cdots]\bigotimes\mathbb{Z}_p[y_0,y_1,y_2,\cdots] &\quad\mbox{for}\quad p>2,\\[1mm] \mathbb{Z}_2[x_0,x_1,x_2,\dots]&\quad\mbox{for}\quad p=2, \end{cases} $$ where $\deg(x_r)=2p^r-1,\,\deg(y_r)=2p^{r+1}-2$. In particular \begin{eqnarray*} \sum_{q=0}^\infty\dim H_q\big(\Omega^2S^3;\mathbb{Z}_p\big)\,t^q&=& \prod_{r=0}^\infty\frac{1+t^{2p^r-1}}{1-t^{2p^{r+1}-2}}\qquad\text{for $p>2$},\\ \sum_{q=0}^\infty\dim H_q\big(\Omega^2S^3;\mathbb{Z}_2\big)\,t^q&=& \prod_{r=0}^\infty\frac{1}{1-t^{2^{r+1}-1}}\,. \end{eqnarray*}

Theorem 2. There is an isomorphism of $\mathbb{Z}$--modules $$ H_q(\Omega^2S^3;\mathbb{Z})= \begin{cases} \mathbb{Z}&\text{for $q=0,1$},\\[1mm] \bigoplus_{p\geqslant 2}\beta_p(H_{q+1}(\Omega^2S^3;\mathbb{Z}_p))&\text{for $q\geqslant 2$}, \end{cases} $$ where $\beta_p:H_{q+1}(\Omega^2S^3;\mathbb{Z}_p)\longrightarrow H_q(\Omega^2S^3;\mathbb{Z})$ is the Bockstein homomorphism corresponding to the exact sequence of coefficients $0\longrightarrow\mathbb{Z}\stackrel{\times p}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{Z}_p\longrightarrow 0$ for prime $p\geqslant 2$. Homomorphisms $\beta_p$ are the graded injective differentiations. The action of $\beta_p$ is defined by the formulas $$ \beta_p(x_0)=0,\qquad\beta_2(x_r)=x^2_{r-1},\qquad \begin{cases} \beta_p(x_r)=y_{r-1},\\ \beta_p(y_r)=0 \end{cases} \quad\text{for $p>2$}, $$ where $r\geqslant 1$.