Adams Cobar Equivalence

Let F be the homotopy fibre of a continuous map Y --> omega X, with X simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA's and DGA modules, OMEGA-C*(X) --> congruent-to CU*(OMEGA-X), and OMEGA(C*omega(Y); C*(X)) --> congruent-to CU*(F), where on the left-hand side OMEGA(-) denotes the cobar construction. Our equivalences are natural in X and omega. Using this result we show how to read off the algebra H*(OMEGA-X; R) and the H*(OMEGA-X; R) module, H*(F; R), from free models for the singular cochain algebras CS*(X) and CS*(Y); here we assume R is a principal ideal domain and X and Y are of finite R type.