Acyclic models and fibre spaces

Introduction. Most of the recent rapid development of the homology or cohomology theory of fibre spaces is based on the use of spectral sequences, as introduced by J. Leray [1]. In order to use spectral sequences successfully, it is almost always necessary to have a good deal of information about the term usually called for cohomology E2 or for homology E2 of the spectral sequence. In his original work, Leray proved that the term E2 in the spectral sequence for the Cech cohomology of a fibre space is naturally isomorphic to the cohomology of the base space with local coefficients in the cohomology of the fibre. The appropriate analogues of this result for cubical singular homology, and cohomology were proved by J.-P. Serre [3 ], and are fundamental in applications of homology theory to homotopy theory. The main object of this paper is to calculate the term E2 in the singular homology spectral sequence of a fibre space by the use of the theory of acyclic models of Eilenberg and MacLane [4]. Notice that we said the singular homology spectral sequence of a fibre space, and did not specify whether we meant simplicial or cubical singular theory. One of our results asserts essentially that there is a singular homology spectral sequence of a fibre space, and that it may be obtained by using either simplicial or cubical singular theory. Further, we show that in cubical singular homology theory a filtration somewhat different from the one used by Serre may be used to obtain the spectral sequence. This filtration is symmetric in all coordinates, and consequently much more convenient to use when dealing with fibre spaces wherein a multiplication is defined in the base space and total space, and such that projection map is a homomorphism. This fact will be exploited elsewhere. In the course of our work, it has been necessary to make a fairly extensive axiomatic investigation of the notion of a "singular theory," and of the notion of "local coefficients." One outcome of this work is a proof of the fact that cubical singular homology and simplicial singular homology coincide even with local coefficients. For ordinary coefficients this was proved by Eilenberg and MacLane [4]. Since much of our work is quite abstract, the general theory will be interspersed with examples from cubical singular theory. Technically this paper is divided into chapters, and eachi chapter into sections. The notation 11.3.4 refers to the 4th numbered statement in the