On the Künneth theorem

Abstract

Introduction. The Kiinneth theorem asserts that the homology H(X0 Y) of the product of two free abelian positively graded chain complexes is isomorphic to Tor(HX, HY) = HX 0HY+Tor1(HX, HY). This isomorphism however gives no insight into the effect of chain maps; that is to say, H(f 0g) cannot be computed in terms of Hf and Hg. Alternatively the difficulty may be expressed by saying that the isomorphism in question is not canonical. It appears that in order to produce a functorial Kiinneth theorem some additional invariant of a chain complex is needed, and indeed Bockstein [I I and Palermo [4] have produced such theorems using as the starting point the homology or cohomology spectra. In this paper we adopt the alternate course of introducing as an invariant the affine space of orientations (?2) of a chain complex. Having defined the notion of skew-isomorph (?3) of a group with respect to such an affine space we then see that the homology of the product, as well as cohomology groups, groups of homotopy classes and so forth, are just skew-isomorphs of those produced by the usual functors of homological algebra. In ?6 we use the same device to sharpen the notion of derived functor, and with these sharpened functors we are able to compute the space of orientations of a product complex (?7). There is an obvious application, which however we omit, to the triple (etc.) product studied by MacLane [3]. The external product in cohomology, and also the cup-product, have straightforward expressions in the language here adduced. These are given in ?8. 1. Unfolding chain complexes. By chain complexes we shall mean free abelian graded groups with derivations of degree -1. We shall further require that these groups be graded by nonnegative degrees, i.e., that their homogeneous components of negative degree be 0. Some of our results remain valid without the latter restriction; the reader may easily supply the generalizations for himself. We shall also have occasion to consider bicomplexes, by which we understand free abelian bigraded groups A = E,,oo A,q with derivations of degree (0, 1). We shall call p the principal, q the resolvent degree of Apq. The notions of chain map and chain homotopy of chain complexes are the usual ones; we extend them by analogy to bicomplexes. Observe that in the latter case a chain homotopy of degree (p, q) connects maps of degree (p, q1).