Summer Term 2014 - BMS Advanced Course
Welcome to the website! The lecture is taught by Pavle Blagojević and Holger Reich and is a continuation of Topologie I. Although it is helpful to have taken Topologie I, it is not necessary.
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).
Contact
Contact | Office Hours: | ||
Lecture | Pavle Blagojević and Holger Reich | blagojevic(at)math.fu-berlin.de, holger.reich(at)fu-berlin.de | TBA |
Tutorial | Albert Haase | a.haase(at)fu-berlin.de | Mon 14-15, 002, Arnimallee 2 |
Lectures
Lectures | ||
Wed | 10:15 - 11:45 | HS 001, Arnimallee 3 |
12:15 - 13:45 |
Tutorials and Problems
Tutorials | ||
Mon | 14:15 - 15:45 | SR 031, Arnimallee 6 |
In the tutorial, we will learn a few new things, expand on topics from class, and occasionally review exercises.
Every week exercises 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 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 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.
Problem Sheets