Homology With Local Coefficients

Abstract

In a recent paper [16] the author has had occasion to introduce and use what he believed to be a new type of homology theory, and he named it homology with local coefficients. It proved to be the natural and full generalization of the Whitney notion of locally isomorphic complexes [18]. Whitney, in turn, credits the source of his idea to de Rham's homology groups of the second kind in a nonorientable manifold [13]. It has since come to the author's attention that homology with local coefficients is equivalent in a complex to Reidemeister's Uberdeckung [10]. Since this new homology theory (which includes the old) seems to have such wide applicability, a complete review of the older theory is needed to determine to what extent and in what form its theorems generalize. The object of this paper is to make such a survey. The general conclusion is that all major parts of the older theory do extend to the new. In addition the newer theory fills in several gaps in the old. The most noteworthy of these is a full duality and intersection theory in a non-orientable manifold (?14). For the sake of completeness, some of the results of Reidemeister have been included. The new approach and new definitions make for easier and more intuitive proofs. They lead also to results not obtained by Reidemeister. The most important is a proof of the topological invariance of all the homology groups obtained.' In addition developments are given of the subjects of multiplications of cycles and cocycles, chain mappings, continuous cycles, and Cech cycles. Part I contains an abstract development of systems of local groups in a space entirely apart from their applications to homology. Any fibre bundle over a base space R [18] determines many such systems in R (one for each homology group, homotopy group, etc., of the fibre). These are invariants of the bundle. They should prove to be of some help in classifying fibre bundles. Part II, which contains the extended homology theory, presupposes on the part of the reader a knowledge of the classical theory such as can be found in the books of Lefschetz [7] and Alexandroff-Hopf [1].