Spectral sequence operations converge to Araki-Kudo operations (Q387396)

Let \(\mathcal{C}\) be a fixed \(E_{\infty}\)-operad, let \(X\) be a cosimplicial \(\mathcal{C}\)-space and let \((E^{r}_{-s, t})\) be the mod. \(2\) homology spectral sequence associated to \(X\).NEWLINENEWLINEIn [J. Pure Appl. Algebra 217, No. 7, 1350--1377 (2013; Zbl 1277.55009)], the author gave a new proof that there are operations NEWLINE\[NEWLINEQ^{m}: E^{\infty}_{-s, t} \rightarrow E^{\infty}_{-\nu, \nu - s -m + t}NEWLINE\]NEWLINE with NEWLINE\[NEWLINE\nu = \begin{cases} s, & \text{if \(m \geq t\);} \\ t - s - m, & \text{if \(t-s \leq m \leq t\).} \end{cases}NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, the author compares these operations \(Q^{m}\) with the usual Araki-Kudo operations in the mod. \(2\) homology \(H_{*}(\text{Tot}(X))\) of the total space of \(X\) (\(\text{Tot}(X)\) is in this context a \(\mathcal{C}\)-space). For this, he uses Bousfield's identification of \(H_{*}(\text{Tot}(X))\) as the abutment of the homology spectral sequence associated to \(X\).