Homology of Fibre Spaces

Abstract

We conclude this volume with an introduction to the method of spectral sequences for studying the homology of a fibration. In §1 we consider a filtered pair (X, A). The homology groups of the triples (X _p , X_p-1, A) are linked together in an intricate way. They can, however, be assembled into two graded groups which are connected by an exact triangle Such a diagram is called an exact couple; the notion is due to Massey [1; 2]. The basic operation on exact couples, that of derivation, gives rise to a new exact couple Hence the process can be iterated to obtain an infinite sequence of diagrams.