Topology II

elective courses

Topology I 1000-113aTP1a

The course starts from the notion of a fundamental group of a space and its relations to the theory of covering spaces. The second part is devoted to a brief introduction to singular homology theory of topological spaces. The last few lectures should present important applications of previously introduced concepts.

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).

G. Bredon, Topology and Geometry. Graduate Texts in Mathematics 139, Springer-Verlag, New York 1993.

K. Janich, Topology. Springer 1984.

W. Massey, A Basic Course in Algebraic Topology. New York, 1991