LIFTING T-STRUCTURES AND THEIR DUALS

We de¯ne and study a concept of Tf -space for a map, which is a generalized one of a T-space, in terms of the Gottlieb set for a map. We show that X is a Tf -space if and only if G(§B; A; f;X) = [§B;X] for any space B. For a principal ¯- bration Ek ! X induced by k : X ! X0 from ² : PX0 ! X0, we obtain a su±cient condition to having a lifting T ¹ f -structure on Ek of a Tf -structure on X . Also, we de¯ne and study a concept of co-Tg-space for a map, which is a dual one of Tf -space for a map. We obtain a dual result for a principal co¯bration ir : X ! Cr induced by r : X0 ! X from ¶ : X0 ! cX0.