Localitzacions homològiques i homotòpiques

This is an interdisciplinary project in topology and algebra, focusing on the study of continuous maps which induce isomorphisms in a given homology theory. Localizations are natural transformations defined on topological spaces, which convert homology equivalences into homotopy equivalences in a universal way. Recently developed techniques have explained much more clearly the effect of homological localizations, as well as the effect of other more general transformations, called homotopical localizations. In this project, several approaches are proposed towards the solution of open problems in the theory of homotopical localization. In particular, we intend to determine which algebraic structures are preserved under the effect of localizations. For this purpose, analogous transformations will be studied in the category of groups and the homotopy category of simplicial groups. One of the major applications could lead to a generalization of classical theorems about homotopy groups of finite cell complexes