Singular homology and cohomology with local coefficients and duality for manifolds
This paper presents a foundational treatment of the topics in the title. The results are the standard ones, but are obtained in the context of a carefully devised and consistent system of definitions. The whole treatment concerns theories defined on pairs of subspaces of a fixed space \(X\), satisfying carefully formulated axioms (including tautness -- here renamed continuity -- in the case of cohomology). Initially, cohomology theories are defined on closed pairs, homology on open pairs, but this is widened using a notion of pretheories. Since the author is concerned with singular theory, these are accompanied by notions of chain and cochain functors. Singular theories satisfying the axioms are constructed using a local coefficient system and come in two flavours: for each of homology and cohomology, with compact or with closed supports. The details, and the corresponding additivity axioms are given carefully in the paper and lead up to a proof of the expected duality theorems. One strange feature of the treatment is that for a number of results the local system has to be supposed of the form \(\Hom(\Gamma, G)\) (\(\Gamma\) a coefficient system, \(G\) a group).
scientific article; zbMATH DE number 572471