Lectures

Mi 10-12 HG D 3.2

Fr 13-15 HG D 3.2

This is an introductory course in algebraic topology. Topics covered include:

• singular homology
• cell complexes and cellular homology
• the Eilenberg-Steenrod axioms
• cohomology

Along the way we will introduce the basics of homological algebra and category theory.

Prerequisites

You should know the basics of point-set topology.

Useful to have is a basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Students not familiar with this topic can look this up, for example in Chapter 3, Section 1-6 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Members of ETH can legally download the ebook trough the ETH network.)

Some knowledge of differential geometry and differential topology is useful but not necessary.

Some (elementary) group theory and algebra will also be needed.

Lectures Content

Week	Content
Week 1	Homotopy- definitions, basic constructions and properties
Week 2	Singular homology - definition, basic properties, some simple calculations. Hurewicz theorem on the 1'st homology group
Week 3	Chain complexes, exact sequences, the homology long exact sequence
Week 4	Relative homology, reduced homology, axiomatic approach to homology, the homology of the sphere and related calculations
Week 5	Degree of maps, calculation of degree, applications
Week 6	CW-complexes
Week 7	Cellular homology
Week 8	Cellular approximation theorem
Week 9	Applications of cellular homology, Euler characteristic.
Week 10	Chain homotopies, cross product, proof of the homotopy axiom
Week 11	Preparation for the proof of excision axiom
Week 12	Barycentric subdivision, proof of the excision axiom, the Mayer-Vietoris long exact sequence
Week 13	The Mayer-Vietoris long exact sequence (cont.)
Week 14	Further applications of homology theory. The Jordan separation theorems, invariance of domain, invariance of dimension

The new exercises will be posted here.

If you have any questions concerning the exercises, please don't hesitate to contact Berit Singer .

Exercise sheets	due by	Solutions
Exercise sheet 1	Friday September 28	Solutions 1
Exercise sheet 2	Friday October 19	Solutions 2
Exercise sheet 3	Friday November 9	Solutions 3
Exercise sheet 4	Friday November 30	Solutions 4
Exercise sheet 5	Friday December 20	Solutions 5

Old Exams