I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/PUBLICATIONS/mapping-class-groups-and-function-spaces.pdf.
From now on coefficients of (co)homology will be taken in $\mathbb{F}_p$ with $p$ an odd prime.
The authors are studying the mod p spectral sequence associated to the fibration $C(S^2;S^{2q})\hookrightarrow E(\eta ;S^{2q})\to BSO(3)$, where $E(\eta ;S^{2q})$ are the fiberwise configurations with label in $S^{2q}$ associated to the fibration $S^2\hookrightarrow BSO(2)\to BSO(3)$. They want to show that some classes $\lambda_j\in H^*(C(S^2;S^{2q}))$ are infinite cycles (I have included a more detailed description of these classes in the appendix at the end of the question).
As far as I understand the trick is the following:
- Consider the composite $j:C(S^2;S^{2q})\to E(\eta; S^{2q})\to \Omega^{\infty}\Sigma^{\infty}(\bigvee_{k\geq 1}D_k(\eta;S^{2q}))$ where the first arrow is the obvious inclusion and the third arrow is the total Hopf invariant.
- "Standard computations" give that $\lambda_j$ are in the image of $j^{\ast}$, therefore they must be restriction of classes in $H^*(E(\eta;S^{2q}))$. It follows that they are infinite cycles.
Question: The point I am missing is why point $(2)$ is true. In the paper they omit the computation, and I am not able to fill the missing details. Any help in understanding this point will be very useful, as well as some references that could help to understanding it (maybe the computation is indeed standar but I am missing some theory to do it).
Appendix The classes $\lambda_j$, ${j\geq 0}$ are described as follows in Section 8 of the paper, when the authors compute the cohomology of the fiber $C(S^2;S^{2q})$: The trick is to observe that this space is homotopy equivalent to $\Lambda^2S^{2q+2}$ and then they do a computation of the spectral sequence of the fibration $\Omega^2S^{2q+2}\hookrightarrow \Lambda^2S^{2q+2}\to S^{2q+2}$. The second page of this spectral sequence is $E_2=H^{\ast}(S^{2q+2})\otimes H^{\ast}(\Omega S^{2q+1})\otimes H^{\ast}(\Omega^2 S^{4q+3})$.
The classes $\lambda_j$ are defined as generators of $Ker(H^{\ast}(\Omega^2S^{4q+3})\to H^{\ast}(S^{4q+1}))$ as an algebra over the Steenrod algebra. More concretely, if $x=[S^{4q+1}]$ and $\beta Q_1 x$ is the Bockstein of $Q_1 x$ ($Q_1$ is the first Dyer Lashof operation), then $\lambda_j$ should be the duals of $(\beta Q_1 x)^{p^j}$, but I am not completely sure about this. In any case, after some computations (Lemma 8.2) they found out that these classes are infinite cycles, so they can be seen as classes of $H^{\ast}(\Lambda^2S^{2q+2})=H^{\ast}(C(S^2;S^{2q}))$.