Cohomology and homology theories for categories of principal 𝐺-bundles

Abstract

Introduction. In [4] G. W. Whitehead gives a method by which all extraordinary homology and cohomology theories may be generated by fundamental homotopy constructions. Unfortunately the definition of the extraordinary theories used is not sufficiently general so as to include all theories that would normally be considered as (co) homology theories. An example of a theory not included is (co) homology with local coefficients. In this paper, we will show how to extend the methods of Whitehead so as to include such theories. Let G be a fixed continuous group. If we consider the category of principal G-bundles over C.W. complexes, we may define a connected sequence of functors from this category to the category of abelian groups, satisfying the usual axioms of extraordinary (co) homology theory, with principal G-bundles replacing spaces, and bundle maps and homotopies replacing maps and homotopies. It is the purpose of this paper to show that such theories may be generated by homotopy constructions. In particular, we will indicate how local coefficient theory may be generated in this manner. In order to develop the homology theories spoken of above, it will first be necessary to define a class of spaces closely related to C.W. complexes but with just enough additional generality so as to include spaces for which a useful C.W. decomposition may not be available, but which are structured enough to give us a good deal of the type of information usually associated with a C.W.-decomposition. The definition of cohomology groups used in this paper was proposed by Professor Israel Berstein of Cornell University and will be exploited by him in a forthcoming paper on the Thom Isomorphism. This paper represents part of a doctoral dissertation written under Professor Berstein at Cornell University. 1. Piecewise-C.W. complexes. We will need three lemmas from homotopy theory. The first two will not be proved as they are quite easy. In the following,