Path-loop fibration

The evaluation map from the based path space gives a fiber sequenceSince $PX$ is contractible, it relates invariants of $X$ to those of its loop space.

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.