Cap pairings and spectral sequences (Q949639)

The author shows that for a pointed cosimplicial space \(X\) and pointed simplicial spaces \(A\) and \(B\), each cap pairing \(X^{n} \wedge A_{n} \to B_{n}\) satisfying the usual projection formula induces a pairing \( \operatorname{Tot} X \wedge\operatorname{Diag} A \to\operatorname{Diag} B\). This cap pairing induces a cap pairing of homotopy spectral sequences. That is, there is an induced pairing, at least in positive dimensions, of homotopy spectral sequences \(E_{r}^{p,q}(X) \otimes E^{r}_{s,t}(A)\to E^{r}_{ s-p,q+t }(B)\) which is compatible with abutments in the sense that \(F^{p}\pi_{q} \operatorname{Tot} X\otimes F_{s}\pi_{t}\operatorname{Diag} A \to F_{s-p}\pi_{q+t}\operatorname{Diag} B\), exploiting that the class of cap pairings going out of \(X\) and \(A\) factor through a universal target.