Cocyclic morphisms and dual G-sequences

In this paper, we first show the monomorphicity of a map implies exactness of G-sequence of the map relative to any space. Next we introduce the concept of cocyclic morphisms in the category of pairs and certain homotopy invariants. We also study its properties, especially, the duality between cyclic morphisms and cocyclic morphisms. Finally we introduce the dual G-sequences in the category of pairs as the subsequence of the cohomology sequence and show a condition for the dual G-sequence to be exact.