Twisted Tensor Models for Fibrations

Let k be any field and let k-0-dga(*) be the category of connected cochain k-algebras. We define the notion of a twisted extension in this category, which generalizes the classical notion of a KS-extension. A morphism in k-0-dga(*) is called an algebraic fibration if it has the weak homotopy type of a twisted extension. An algebraic fibration may possess ''fibers'' with different cohomology algebras, though all with the same cohomology as vector space. Our main result is that, for any fibration F --> E --> B between 1-connected topological spaces of finite type, the induced map on the (reduced) cochain algebras C-*(B; k) --> C-*(E; k) is an algebraic fibration, and that one of the algebraic fibers has the weak homotopy type of C-*(F; k).