Singular homology and cohomology

3-4 hours per week.

Lecturer: Marcel Bökstedt

Content:
Singular homology and cohomology is an algebraic and combinatorial method of constructing invariants for topological spaces. This is in contrast with the deRham cohomology treated in Topologi 1 where one uses differential geometry to obtain closely related topological invariants. The singular version of homology is basic in most parts of algebraic topology. A main theme in the course is the interplay between the algebraic constructions of 'homological algebra' and the geometric implementations.

Topics:

The fundamental group. Covering spaces. Singular homology. The axioms of homology theory and their verification for singular homology. Cohomology. The Universal Coefficient Theorems. Products. Duality Theorems.

Prerequisites: A basic knowledge of point set topology and of algebra should be enough.

Literature: Alan Hatcher, Algebraic Topology, Cambridge University Press. The book is supposed to appear in the spring 2001. It is (and will continue to be) available from the net, on

www.math.cornell.edu/hatcher/