On The Sectional Fibrations

Given a fibration Fâ†'Eâ†'B, we wonder whether can p admit a section σ such that pa=id. In this paper, we treat rational sections and transform the problem to some realization problems under different restrictions. Further, some general criterion is also developed using the derivations of the Sullivan models of the fibre F.In addition, we also consider the special case when H*(B) is generated by one generator or B is simply the wedge of such spaces. It turns out that the problem is equivalent to the realizability of some certain commutative differential graded algebra constructed from the minimal model of p. Then the space of the sections of p is de-scribed by the Maurer-Cartan elements of some L∞algebra as set which also preserves homotopy relation.Finally, we also develop a Lie model for the space of the sections of p using the localization method, which helps us to consider an algebraic realization problem in the context of Lie algebras.