The homotopy category of exact couples

Abstract

THE CATEGORY of Cl&‘-complexes and stable homotopy classes of continuous maps has attracted the attention of topologists for more than a decade (see [l], [2], [3], [Sb] or [9]). As a category, it has many unusual properties: it is not Abelian, but sums agree with prohicts: every object is projective, etc. Puppe [Sb] has studied a general class of stable homotopy categories. Nevertheless, we have at present no general structure theorem for this category. Of course, a full algebraic description cannot be expected at this time, for such a description would describe all classes of maps .Y -+ Y. It appears to me that, for the present, an ideal description would consist in some algebraic structure, defined in terms of the stable homotopy ring G, (see [IO]). We aim for the following here: (A) We construct ($1) a homotopy theory in the category of exact couples of graded modules over a ring R. (B) We construct a functor .F (92) from finite CL+‘-complexes to exact couples, which respects the notions of homotopy, and moduio maps on which .F vanishes, we show (93) that _F imbeds the stable homotopy category into the quotient, under homotopy, of the category of exact couples of G,-modules. One may then heuristically regard the exact couples as a generalization of the stable homotopy category. (C) In $3, it is shown that one may define homotopy groups, homology and cohomology groups, and the Hurewicz homomorphism in the category of exact couples over any graded ring. We establish the homotopy invariance of these notions, as well as compatibility with the usual definitions, via the functor 9, in the case of the ring G, . In $5, we show that the homotopy groups of an exact couple form a representable functor. $G shows how the classical Theorems of Hurewicz and Whitehead hold in the category of exact couples. (D) In $7, we consider some examples and show that the analogue of a conjecture of [3] is false in the category of exact couples over a graded polynomial ring. $8 contains some remarks as to why the obvious method of constructing cones does not work in the category of exact couples. One virtue of the category of exact couples of R-modules is that it adds a new dimension to stable topology by permitting the rin g, classically G, , to vary. A more practical advantage