Homology and cohomology of diagrams of topological spaces

Abstract

Homology and cohomology of objects other than ordinary topological spaces have been investigated by several authors. Let X be a G-space and ${\cal F}$ be a family of all closed subgroups of G. The equivariant cellular structures are obtained by attaching n-cells of the form G/H $\times$ B$\sp{\rm n}$, where B$\sp{\rm n}$ is the unit n-ball and H $\in\ {\cal F}$. A construction of the equivariant singular homology and cohomology for X and ${\cal F}$ was given by Soren Illman. In this thesis ${\cal F}$ is replaced by a small topological category and the G-space X is replaced by a functor taking values in k-spaces and called a diagram of topological spaces. The cellular structures for diagrams are obtained by attaching cells D$\sb{j}\times\rm B\sp{n}$ where D$\sb{j}$ is a representable functor. The purpose of this thesis is to investigate homology and cohomology for diagrams. We review the foundation of homotopy theory and cellular theory for diagrams. We propose axioms for homology and cohomology modeled on Eilenberg-Steenrod system of axioms and prove that they, as in the classical case, determine uniquely homology and cohomology for finite cellular diagrams. We give the generalization of Illman's equivariant singular homology and cohomology to diagrams of topological spaces and we prove that this generalization satisfies all introduced axioms. Also we prove the comparison theorem between the sheaf cohomology for diagrams developed by Robert J. Piacenza and the singular cohomology for diagrams developed in this thesis.