### Rational fibrations in differential homological algebra.

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In this abstract we present an explicit formula for a cycle representing the top class of certain elliptic spaces, including the homogeneous spaces. For thet, we shall rely on the connection between Sullivan's theory of minimal models and Rational homotopy theory for which [3], [6] and [10] are standard references.

Via the Bousfield-Gugenheim realization functor, and starting from the Brown-Szczarba model of a function space, we give a functorial framework to describe basic objects and maps concerning the rational homotopy type of function spaces and its path components.

In this paper we find a formula for the rational LS-category of certain elliptic spaces which generalizes or complements previous work of the subject. This formula is given in terms of the minimal model of the space.

In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite complex which acts nilpotently on the homotopy groups is itself nilpotent.

In this paper we present an approximation to the de Rham theorem for simplicial sets with any coefficients based, using simplicial techniques, on Poincaré's lemma and q-extendability.

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