ABSTRACT

Homological algebra is an example of a bidirectional interaction between algebra and homotopy theory, especially group cohomology since it can be computed from the category of modules over the group ring or just as singular cohomology of the classifying space of the group. This algebraic nature of classifying spaces produced spectacular results in comparing algebraic and homotopical constructions. Stable homotopy theory of saturated fusion systems deserves a special section since the developments in this area have been ahead of the unstable theory in solving relevant problems in the theory, like the existence of a classifying space or the functoriality of the classifying space construction. The theory of fusion systems is a new way to solve questions in finite group theory and homotopy theory involving conjugacy relations. The homotopy theory of maps between classifying spaces is less developed, as well as the stable homotopy theory.