Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to $PH_*(QY)$, and let's say $Y$ itself is a suspension (as in the other question, this extra hypothesis is not crucial). By the quoted question,, all elements in the kernel are either in the image of $Q_0$, hence degree divisible by $2p$, or in the image of $\beta Q_1$, hence degree congruent to $(-2)$ modulo $2p$.
This latter fact can be seen easily using the Eilenberg-Moore spectral sequence for the path-loop fibration over $Q\Sigma Y$: $QY\to pt \to Q\Sigma Y$. The $E_2$ term looks like $$Cotor _{H_*(Q\Sigma Y;Z/p)}(Z/p,Z/p)\cong (Tor _{H^*(Q\Sigma Y;Z/p)}(Z/p,Z/p))^*.$$
However, by the computation of Dyer-Lashof, $H_*(Q\Sigma Y;Z/p)$ is of the form $$H_*(Q\Sigma Y;Z/p)\cong P[x_i;i\in I]\otimes \Lambda (y_j;j\in J)$$ with all generators $x_i$'s and $y_j$'s primitive. Thus by dualizing $$H^*(Q\Sigma Y;Z/p)\cong TP_1[(x_i^{p^m})^*;i\in I,m\in \mathbf{N}]\otimes \Lambda (y_j^*;j\in J)$$. Therefore, the Tor group looks like $$Tor _{H^*(Q\Sigma Y;Z/p)}(Z/p,Z/p)\cong \Lambda (\sigma ^{-1}((x_i^{p^m})^*))\otimes \Gamma (\tau (x_i^{p^m})^*)) \otimes \Gamma (\sigma ^{-1}(y_j^*))$$ where $\tau $ denotes the transpotence. All exterior generators have to survive to $E_{\infty}$ as all p-th powers in $H_*(Q\Sigma ^Y;Z/p)$ are known to be in the image of the homology suspension. SInce everything else is concentrated in even degrees, the spectral sequence collapses at $E_2$, and in homology, by dualizing we get
$$E_2=E_{\infty}\cong \Lambda (\sigma ^{-1}((x_i^{p^m})))\otimes P (\tau (x_i^{p^m})^*)) \otimes P (\sigma ^{-1}(y_j^*)).$$ Thus the only polynomial generators that suspends trivially correspond to the transpotence elements, having the internal degree divisible by $2p$ and homoogical degree $2$, so the total degree congruent to $(-2)$ mod $p$. If we also consider "decomposable primitives", as there are also $p$-th powers, we arrive at the conclusion above.
My questions, now, are :
- Is there any reference for the Eilenberg-Moore spectral sequence for the path-loop fibration over $Q\Sigma X$ at odd primes?
- Is there any reference for the degrees of elements in the kernel of the suspension map for $QX$ (again, at odd primes)?
