Course Description

This course is a continuation of Topologie I, taught last semester. Although it is helpful to have taken Topologie I to follow the present course, it is not (absolutely) necessary. The aim of Topologie II is a skillful handling and thorough understanding of all notions of homology and cohomology, the cup and cross product, as well as several types of duality. The course will roughly be structured as follows:

• Categories and functors, chain complexes

• Singular homology, chain homotopy

• Mayer-Vietoris Sequence, Jordan Curve Theorem

• Reduced homology, relative homology, Alexander's theorem

• Simplicial homology

• Degrees, Euler characteristic, Lefschetz number, Lefschetz fixed point theorem

• CW complexes

• Cellular homology

• Eilenberg–Steenrod axioms

• Künneth Theorem

• Universal Coefficient Theorem

• Singular cohomology, simplicial cohomology

• Cup product

• Cross product, topological manifolds

• Poincaré Duality

• Alexander Duality

• Manifolds with boundary

The course will loosely be based on lecture notes by Milnor (not freely available). We also recommend the books by J. Munkres ("Elements of Algebraic Topology", Addison-Wesley 1984) and A. Hatcher ("Algebraic Topology", Cambridge U Press 2002, also online) and the succinct lecture notes by J. P. May ("A Concise Course in Algebraic Topology", online).