Integral homology of braid groups as a ring

Question

Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each $i$ and $k$. There is a ring structure on $\bigoplus_{k,i} H_i(Br_k;\mathbb Z)$ induced by the maps $Br_k \times Br_j \to Br_{k+j}$. Is the ring structure on $\bigoplus_{k,i} H_i(Br_k;\mathbb Z)$ known? Cohen-Lada-May compute it with field coefficients. Equivalently, is the ring structure on $H_*(\Omega^2 S^2;\mathbb Z)$ known?

Answer 1

Using the Hopf fibration you can show that $\Omega^2S^2\simeq\mathbb{Z}\times\Omega^2S^3$. Also $\pi_1(\Omega^2S^3)=\pi_3(S^3)=\mathbb{Z}$, so the Universal Coefficient Theorem gives $[\Omega^2S^3,S^1]=H^1(\Omega^2S^3)=\text{Hom}(\pi_1(\Omega^2S^3),\mathbb{Z})$. Using this we see that $\Omega^2S^3=S^1\times W$ for some $W$. We then find that the homotopy groups of $W$ are all finite, and thus that the homology groups of $W$ are all finite. Thus, the reduced integral homology of $W$ is the direct sum of the $p$-local homologies for all $p$, and it is easy to see that the product of any two summands is zero in the ring structure. The Cohen-Lada-May calculation shows that $H_*(W;\mathbb{Z}/p)=P\otimes E$, where $E$ is exterior on generators $x_k$ of degree $2p^k-1$ for $k>0$, and $P$ is polynomial on generators $y_k=\beta(x_k)$ of degree $2p^k-2$. It follows that the Bockstein homology $\ker(\beta)/\text{image}(\beta)$ is just $\mathbb{Z}/p$ in degree zero (by decomposing the whole ring as the tensor product of subrings generated by a single pair $\{x_k,y_k\}$, and applying the Kunneth isomorphism). This means that in positive degrees we have $\ker(\beta)=\text{image}(\beta)$. Let $\rho\colon H_*(X;\mathbb{Z})\to H_*(X;\mathbb{Z}/p)$ be the reduction map. There is a lifted Bockstein operation $\widetilde{\beta}\colon H_k(X;\mathbb{Z}/p)\to H_{k-1}(X;\mathbb{Z}_{(p)})$ with $\beta=\rho\widetilde{\beta}$ and $\widetilde{\beta}\rho=0$. Using this one can check that $\rho$ gives an isomorphism $H_*(W;\mathbb{Z}_{(p)})\to\ker(\beta)$ in positive degrees. I don't think you can expect to get a nice presentation of this ring by generators and relations.

In this kind of context, you should usually avoid integral homology if you possibly can. Most things that you might want to do with integral homology can be done by combining rational and mod $p$ homology. In some cases Brown-Peterson homology is more tractable than integral homology.