Applications of perturbation theory to iterated fibrations
The purpose of this paper is to prove the following result. Given a tower of fibrations \[ X=E_ n\to E_{n-1}\to...\to E_ 0\to E_{-1}=* \] in which all of the fibrations \(F_ i\to E_ i\to E_{i-1}\) have simply connected torsion free homology \(H_*(F_ i;R)\) over a ring R, there is a differential \(\partial\) and a chain homotopy equivalence \[ f: C_*(X;R)\to M_ X=(H_*(F_ 0)\otimes_{R}H_*(F_ 1)\otimes_{R}...\otimes_{R}H_*(F_ n),\partial). \] Moreover, there is a homotopy associative coalgebra structure on \(M_ X\) such that f extends to a homotopy equivalence \(C_*(\Omega X;R)\to {\tilde \Omega}M_ X\) where \(\Omega\) X is the loop space of X and \({\tilde \Omega}\)M\({}_ X\) is the tilde-cobar construction in the sense of the second author [Trans. Am. Math. Soc. 108, 293-312 (1963; Zbl 0114.394)].
homotopy associative coalgebra structure
scientific article; zbMATH DE number 4028415
scientific article