Transgressive pair

For a fibration $F\to E\to B$, classes $x\in H^r(F)$ and $y\in H^{r+1}(B)$ form a transgressive pair when their images in relative cohomology satisfy $\delta x={\pi }^{\ast }y$. In the Serre spectral sequence this relation is represented by $d_{r+1}x=y$.

Steenrod operations preserve transgressive pairs. With mod-two coefficients, if $(x,y)$ is transgressive, then so is $({\mathrm{Sq}}^{j}x,{\mathrm{Sq}}^{j}y)$ whenever the square is defined by the instability range. The corresponding statement holds for odd-primary reduced powers and the Bockstein, with the appropriate page shift.