WEAK C^f_k-SPACES FOR MAPS AND THEIR DUALS

Abstract

In this paper, we introduce and study the concepts of weak C f k-spaces for maps which are generalized concepts of C f kspaces for maps, and introduce the dual concepts of weak C f k-spaces for maps and obtain some dual results.

G'p-SPACES FOR MAPS AND HOMOLOGY DECOMPOSITIONS

“…The following proposition says that co-H-spaces are completely characterized by the dual Gottlieb sets. A space X is called [22] a co-H p -space for a map p : X → A if there is a map θ : X → X ∨ A such that jθ ∼ (1 × p)∆, where j : X ∨ A → X × A is the inclusion and ∆ : X → X × X is the diagonal map, that is, 1 X : X → X is p-cocyclic. A space X is called a co-T -space [18] if e : X → ΩΣX is cocyclic.…”

Category of Maps and Gottlieb Sets for Maps, and Their Duals

“…In the case, f = 1 X : X → X, g : B → X is called cyclic [15]. We denote the set of all homotopy classes of f -cyclic maps from B to X by G f (B, X) ⊂ [B, X] which is called the Gottlieb set for a map f : A → X.…”

Principal Fibrations and Generalized H-Spaces

“…It is clear that a space X is an H-space if and only if the identity map 1 X of X is cyclic. We called a space X as an H f -space for a map f : A → X [17] if there is a cyclic map f : A → X, that is, there is an H f -structure F : X × A → X such that F j ∼ ∇(1 ∨ f ), where j : X ∨ A → X × A is the inclusion. We showed [17] that if a space X is an H-space, then for any space A and any map f : A → X, X is an H f -space for a map f : A → X, but the converse does not hold.…”

“…For a map f : A → X, there are concepts of H fspaces, T f -spaces, which are generalized ones of H-spaces [17,18].we obtain some sufficient conditions to having liftings Hf -structures and Tf -structures on E k of H f -structures and T f -structures on X respectively. We can also obtain some results about H f -spaces and T f -spaces in Postnikov systems for spaces, which are generalizations of Kahn's result about H-spaces.…”

“…For a map f : A → X, there are concepts of H fspaces, T f -spaces, which are generalized ones of H-spaces [17,18].…”