The generalized dual Gottlieb sets

G'p-SPACES FOR MAPS AND HOMOLOGY DECOMPOSITIONS

“…Clearly any cocyclic map is a p-cocyclic map and also f : X → B is p-cocyclic iff p : X → A is f -cocyclic. The dual Gottlieb set DG(X, p, A; B) for a map p : X → A [20] is the set of all homotopy classes of p-cocyclic maps from X to B. In the case p = 1 X : X → X, we called such a set DG(X, 1, X; B) the dual Gottlieb set [16] denoted DG(X; B), that is, the dual Gottlieb set is exactly same with the dual Gottlieb set for the identity map.…”

“…It is known [20] that for any n, G n (S n × S n ; Z) = G n (S n × S n , p 1 , S n ; Z) = H n (S n × S n ; Z).…”

“…For a map p : X → A, the dual Gottlieb sets G n (X, p, A) of a map p : X → A, which are generalized of dual Gottlieb groups G n (X), are defined in [20]. In general, G n (X) ⊂ G n (X, p, A) ⊂ H n (X) for any map p : X → A.…”

Cocyclic element preserving pair maps and fibrations

“…Any element α ∈ G n (X; F) is called a cocyclic element. Their generalizations were studied by Varadarajan [12], Lim [9], Oda [11] and Yoon [15].…”

“…Let K(F, n) be the EilenbergMacLane space. Yoon [15] introduced a subset G n p (X; F) of H n (X; F) by for any map p : X → A (see Section 2). An element α = [a] ∈ G n p (X; F) is called a p-cocyclic element.…”