Course Description

This We understand this course is a continuation of Topologie I, taught last semesteras a comprehensive beginners course in algebraic topology. 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 for you to thoroughly understand and be able to apply all notions of homology and cohomology, the cup and cross product, as well as several types of dualitysome 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

The course will be loosely 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).

Tutorials and Problems

In the tutorial, we will learn a few new thingssome things that will help us better understand the lecture, expand on topics from class, and occasionally review exercises.

Every week exercises now and then a sheet will appear on this website . It is highly recommended for you to solve the most interesting and most important ones. Solutions to the most important containing exercises, some of which we highly recommend. Do the other exercises if they seem challenging enough or if you don't have an idea of how to solve them immediately. Solutions to select exercises will appear on this website a week later. In order to encourage you to solve exercises, we will (1) periodically ask students to present solutions to an exercise in the tutorial and (2) base some parts of the exam questions on the exercises.

Course requirements are the following: (1) You must actively participate in the course. (2) You must pass an exam at the end of the semester which alone will determine your grade. The date for the exam will be available within the first 2 weeks of class.

. The first written exam will take place on March 3 from 2-4 PM in "Großer Hörsaal Informatikgebäude Takustraße 9". The second written exam will take place on April 13 from 10AM - 12PM in "Seminarraum Animallee 2".

• Hatcher: Idea of Homology 

Exercise Sheets

• Sheet 1 (Oct 15), Sheet 1 Sol
• Sheet 2 (Oct 27), Sheet 2 Sol
• Sheet 3 (Nov 17), Sheet 3 Sol
• Sheet 4 (Nov 26), Sheet 4 Sol
• Sheet 5 (Nov 28),
• Sheet 6 (Dec 22)
• Sheet 7 (Jan 26)
• Sheet 8 (Feb 3)
• Revision (Feb 22)