Algebraic Topology – A Student's Guide: Exact couples in algebraic topology

Introduction The main purpose of this paper is to introduce a new algebraic object into topology. This new algebraic structure is called an exact couple of groups (or of modules, or of vector spaces, etc.). It apparently has many applications to problems of current interest *in topology. In the present paper it is shown how exact couples apply to the following three problems: (a) To determine relations between the homology groups of a space X , the Hurewicz homotopy groups of X , and certain additional topological invariants of X ; (b) To determine relations between the cohomology groups of a space X , the cohomotopy groups of X , and certain additional topological invariants of X ; ( c ) To determine relations between the homology (or cohomology) groups of the base space, the bundle space, and the fibre in a fibre bundle. In each of these problems, the final result is expressed by means of a Leray- Koszul sequence. The notion of a Leray-Koszul sequence (also called a spectral homology sequence or spectral cohomology sequence) has been introduced and exploited by topologists of the French school. It is already apparent as a result of their work that the solution to many important problems of topology is best expressed by means of such a sequence. With the introduction of exact couples, it seems that the list of problems, for which the final answer is expressed by means of a Leray-Koszul sequence, is extended still further.