Cohomology suspension ($\sigma :{\tilde{H}}^q(B;R)\to {\tilde{H}}^{q-1}(\Omega B;R)$)

Pull back along loop evaluation $S^1\times \Omega B\to B$ and take the slant product with the circle's fundamental class. This degree-lowering operation kills positive-degree cup products. It commutes with stable cohomology operations such as Steenrod squares and identifies a transgressing base class with its fiber class in the path-loop fibration.