Topology II 1000-134TP2

Homotopy of maps. Homotopy equivalence. Compact-open topology in function spaces. Homotopy classes as arc components in mapping spaces. The fundamental group of a topological space and its properties - functoriality, dependence on the choice of the base point (2 lectures).

Covering spaces and their morphisms. Lifting of maps and homotopies. Monomorphism of fundamental groups induced by a covering. Group action on a topological space. Regular coverings. Universal covering, existence of a covering with a given fundamental group (draft construction). Classification of coverings over a given space. (4 lectures).

Chain complexes and their homology, chain homotopy. Singular homology of topological spaces, homomorphisms induced by continuous maps. Axioms for homology theory. The Mayer-Vietoris sequence. Computation of homology groups for spheres and surfaces. Examples of applications: non-existence of a retraction of a ball onto a sphere, Brouwer's fixed point theorem, Jordan's closed curve theorem, theorem on region preservation. Hurewicz theorem in dimension 1. (8 lectures).