Winter Term 2014/2015
Welcome to the course website! This course on algebraic topology 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.
Important: lecture hall changed
The lecture hall for Wed, Feb 11, from 10 - 12 AM has been changed to Arnimallee 6 SR 007/008. The second lecture is in the usual lecture hall.
Talks February 4th
- The tutorial next Monday will be postponed to 24 Feb at 10 AM (details of where we meet will follow)
- Here's a list of speakers for Wednesday (11 Feb)
Christoph Horst (Topological quotients, CW complexes)
Arthur Jacot (Degree, properties)
Albert (CW homology, cellular boundary formula)
Reimer Backhaus (Tor and Ext)
Markus Penner (Homology with coefficients, UCT)
Myriam Mahaman (Cohomology, UCT)
Talks are 25 minutes long. If you have questions please email Albert or go to his office hours on Monday.
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).
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 13-14, 002, Arnimallee 2 |
Lectures
Lectures | ||
Wed | 10:15 - 11:45 | HS 001, Arnimallee 3 |
12:15 - 13:45 | SR 031, Arnimallee 6 |
Tutorials and Problems
Tutorials | ||
Mon | 14:15 - 15:45 | SR 031, Arnimallee 6 |
In the tutorial, we will learn some things that will help us better understand the lecture, expand on topics from class, and occasionally review exercises.
Every now and then a sheet will appear on this website 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. 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 parts of the exam 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 of the exam is March 3rd, 2015 from 14-16h in the lecture hall in the Informatikgebäude.
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)