Course Description

We understand this course as a comprehensive beginners course in algebraic topology. Although it is helpful to have taken Topologie I to follow the present course, it is not necessary. The aim of Topologie II is for you to thoroughly understand and be able to apply all notions of homology and cohomology, the cup and cross product, as well as some results on duality (for instance Poincaré duality). The course will roughly be structured as follows (as time permits):

• 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

We 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).