The Leray-Serre spectral sequence provides a method for computing the cohomology of the total space of a fibration from knowledge of the cohomology of the base space and the fibre. By arguing backward through the spectral sequence, the inverse problems of computing the cohomology of the fibre (as for the path-loop fibration) or the cohomology of the base space (as in the case of classifying spaces or Eilenberg-Mac Lane spaces) from the cohomology of the other two spaces in the fibration can sometimes be solved (Theorems 5.16, 6.20, and 6.39). In the particular case of the computation of H* (BG; κ) from H* (G; κ) when G is a compact Lie group, the algebraic relation between H* (G; κ) and H* (BG; κ) is often expressible in the language of homological algebra and derived functors. In pioneering work, [Cartan54], [Moore59], and [Eilenberg-Moore66] developed the correct algebraic setting to explain this relation. We present in this chapter the homological framework that leads to a general method of computation.

To begin, we extend the problem of computing the cohomology of the fibre from the cohomology of the base and total space to a more general question. Suppose π: E → B is a fibration with fibre F and ƒ : X → B is a continuous function.