The main goal of the course is the study of the main properties of topological spaces and a first introduction to Algebraic Topology. At the end of the course, students will be familiar with the basics notions of connection, compactness, separation axioms, and numerability, with the product topology and the quotient topology, with metric spaces, with the theory of homotopy, of the fundamental group and of the coverings. They will also be able to deal with the resolution of problems of a theoretical and practical nature in the field of General Topology and of Algebraic Topology.
In the world of chain complexes E_n-algebras are the analogues of basedn-fold loop spaces in the category of topological spaces. Fresse showed thatoperadic E_n-homology of an E_n-algebra computes the homology of an n-foldalgebraic delooping. The aim of this paper is to construct two spectralsequences for calculating these homology groups and to treat some concreteclasses of examples such as Hochschild cochains, graded polynomial algebras andchains on iterated loop spaces. In characteristic zero we gain anidentification of the summands in Pirashvili's Hodge decomposition of higherorder Hochschild homology in terms of derived functors of indecomposables ofGerstenhaber algebras and as the homology of exterior and symmetric powers ofderived K\\\"ahler differentials. 59ce067264