Topology 539

This is the fall semester of a first year, year long course in algebraic and geometric topology.  The idea is to present the concepts of topology from a historical perspective, starting from a more geometric viewpoint and gradually introducing the algebraic viewpoint. The goal is to keep the geometric picture in the forefront, especially for homology theory.

It would be good for an auditor to have taken Topology and Geometry 530/531 first, although these can be taken concurrently if one knows what a continuous map between compact subsets of Euclidean space means. It is also not inappropriate to have already taken Topology 539, because the perspective now will be more geometric.

The course will in content follow the wonderful book of Allen Hatcher with any additions required by the above goal. I intend to give out exercises and collect them.

The central idea of the course is to show how geometrical ideas lead to Poincare's introduction of the concepts of topology as well as later developments.

Multivalued functions lead to the first homotopy group.

The study of multidimensional integrals lead to Betti numbers. Poincare duality for the Betti numbers lead to triangulations, torsion and homology groups.

Later the invariance question for homology lead to the idea of the induced transformation on homology and a natural transformation between homology theories.

The idea of a geometric cycle will run through the entire development.