T-spaces by the Gottlieb groups and duality

Abstract

It is shown that all the generalized Whitehead products vanish in X and all the components of X TA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.1991 Mathematics subject classification (Amer. Math. Soc): 55P45, 55P35.

Gottlieb groups of function spaces

“…It is easily seen that the property of being a G-space, respectively a Tspace, is inherited by retracts (cf. [16,Cor.2.10] for the latter). Since map(X, Y ; 0) retracts onto Y , the statements in Corollary 5.1 may be replaced by two-way implications.…”

“…Since map(X, Y ; 0) retracts onto Y , the statements in Corollary 5.1 may be replaced by two-way implications. Notice that the first includes [16,Th.2.12]. But the interest in Corollary 5.1 comes from the fact that we are able to deduce properties of the function space map(X, Y ) from properties of Y alone, rather than the other way around.…”

G'p-SPACES FOR MAPS AND HOMOLOGY DECOMPOSITIONS

“…A based map g : X → A is called weakly cocyclic [18] if g * (H n (X)) ⊂ G n (X) for all n. Any cocyclic map is an weakly cocyclic map, but the converse does not holds. It is known [6] that RP 2 is a G -space, but not co-H-space.…”

“…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. The following proposition says that co-T -spaces are completely characterized by the dual Gottlieb sets.…”

Principal Fibrations and Generalized H-Spaces

“…It is easy to show that any H-space is a T -space. However, there are many T -spaces which are not H-spaces in [16]. Let ΣX denotes the reduced suspension of X, and ΩX denotes the based loop space of X.…”