Exact Homomorphism Sequences in Homology Theory

Abstract

The developments of this paper stem from the attempts of one of the authors to deduce relations between homology groups of a complex and homology groups of a complex which is its image under a simplicial map. Certain relations were deduced (see [EP 1] and [EP 2]) which form an extension of the Mayer-Vietoris formulas concerning coverings of a complex by two complexes. In this paper these relations are formalized and are seen to be consequences of the existence of an exact homomorphism sequence (see Definition 3.1). The concept of an exact sequence seems to be due to Hurewicz [WH]. Its principal property is used in the presentation of the Mayer-Vietoris formulas by Alexandroff and Hopf [A-H, pp. 297-299]. It has been used most notably by Eilenberg and Steenrod (reference [E-S]) as one of a very simple system of axioms for homology theory. (See [C] for another use.) In one sense, this paper might have been written from the point of view of exploiting this axiom. Actually we have found the most flexible approach to be the consideration of the exact sequence of homology groups on a Mayer complex as a fundamental algebraic identity. We exploit this identity in two directions. First we obtain a number of duality theorems and second, we investigate chain mappings and some closely related topics on coverings. A considerable part of the paper is methodological in character in that known results are deduced as part of a general line of reasoning. In geometric applications involving Cech homology groups there are two procedures available. One might consider fundamental complexes for a compact metric space, in which case the fundamental algebraic identity could be used, or else one might set up a limiting process. We have used the latter method. Section 1 is concerned with preliminary remarks on notation. In Section 2 we define the term homomorphism sequence and recall the definition of an abstract complex according to Mayer [WM 1, 2, 4] distinguishing chain and cochain complexes notationally and defining homology and cohomology groups. In Section 3 we define the term exact sequence and establish the fundamental construction of the paper, Theorem 3.3. In Section 4, the limit of a direct system of homomorphism sequences is defined and direct limits of exact sequences are shown to be exact. Section 5 gives a summary of needed results on character theory. Compare with papers by Alexandroff [A] and Mayer [WM 4]. These results are used in Section 6 to establish algebraic duality in Mayer complexes. Compare with [WM 4]. The results of Section 5 are used again in Section 7, where limits of inverse systems of homomorphism sequences of compact groups are defined, to show that inverse limits of exact sequences of compact groups are compact. In Section 8, the theory of inverse limits is applied to prove exactness of the homomorphism sequence of homology groups relative to a compact coefficient