The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

Obstructions to the Section Problem in a Fibration with a Weak Formal Base

For a class of Serre fibratations with a weak formal base X (or with a degenerated -algebra structure on the integral cohomology H*(X)), obstructions are defined by means of spherical twisting cochains of . In particular, for a given section on n-skeleton of X, the problem of avoiding the (n+1)th obstruction to the existence of a section on X ⁿ⁺¹ reduces to solving a system of linear equations with respect to cohomology elements of the groups Homotopy classification theorems for sections as well as for weak formal maps are given, too.     

Iterating the Bar Construction

For a 1-connected space X Adams's bar construction B(C*(X)) describes H*(X) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces ⁱ X. In this paper for an n-connected pointed space X a sequence of A()-algebra structures , is constructed, such that for each there exists an isomorphism of graded algebras      

Filtered model of a fibration and rational obstruction theory

For a Serre fibration a filtered model is defined and obstructions to the homotopy equivalence of two fibrations (maps), to the section problem and to the extension problem in rational homotopy theory are given.