Cocyclic element preserving pair maps and fibrations

Some properties of the set of cocyclic element preserving maps are studied for cocyclic elements in cohomology groups defined by Haslam and Yoon. Some subsets of self-homotopy sets and their monoid structures are defined making use of the cocyclic element preserving self maps. Cocyclic element preserving pair maps are examined to obtain further results. Fibration sequences are studied as examples, and non-trivial examples are obtained by making use of Nomura's exact sequence involving groups of self-homotopy equivalences.