The Milnor–Stasheff Filtration on Spaces and Generalized Cyclic Maps

WEAK Cfk-SPACES FOR MAPS AND THEIR DUALS

“…Let f : A → X be a map. A space X is called [6] a C f k -space for a map f : A → X if the inclusion e X k :…”

“…Let f : A → X be a map. A space X is called [6] a C f k -space if the inclusion e X k : P k (ΩX) → X is fcyclic. It is known [6] that a space X is a C f k -space for a map f : A → X if and only if G f (Z, X) = [Z, X] for any space Z with cat Z ≤ k.…”

“…In this paper, we introduce the concepts of weak C f k -spaces for maps which are generalizations of C f k -spaces for maps [6] and study some properties of weak C f k -spaces for maps. We show that a space X is a weak C f k -space for a map f : A → X if and only if W G f (Z, X) = [Z, X] for any space Z with cat Z ≤ k. Let f : A → X and g : B → Y be any maps.…”

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

“…In this paper, we introduce the concepts of W C f k -spaces with respect to spaces which are generalizations of C f k -spaces for maps [5] and study some properties of W C f k -spaces with respect to spaces. We show that for a space Z with mapcat (Z, X) ≤ k, a space X is a W C The LS category of X [3], denoted cat X, is the least integer k such that X is the union of k + 1 open sets U i , each contractible in X.…”

“…Let f : A → X be a map. A space X is called [5] a C f k -space if the inclusion e X k : P k (ΩX) → X is f -cyclic. It is known [5] that a space X is a C f k -space for a map f : A → X if and only if G f (Z, X) = [Z, X] for any space Z with cat Z ≤ k. For any spaces Z, X, we define mapcat (Z, X) ≤ k if for any map g : Z → X, cat g ≤ k. It is known that if cat Z ≤ k, then mapcat (Z, X) ≤ k, but the converse does not hold(see Example 2.6).…”