Welcome to the course website! The lecture This course on algebraic topology is taught by Pavle BlagojevićBlagojević  and Holger Reich and is a continuation of Topologie I. Although it is helpful to have taken Topologie I, taught in the SoSe 2014.it is not necessary.

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 necessaryThis course is a continuation of Topologie I, taught last semester. 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 including products, 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 :

Applications in optimization, number theory, algebra, algebraic geometry, and functional analysis

(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)The course will use material from P. M. Gruber, " Convex and Discrete Geometry" (Springer 2007) and various other sources. There will be brief lecture notes available for course participants with detailed pointers to the literature.

Contact

Lectures

• Lecture Notes Part 1
• Lecture Notes Parts 1-2
• Here's a link to Cynthia Vinzant's (U Mich) article on spectrahedra that was mentioned in the lecture.
• Lecture Notes Parts 1-3
  
• Lecture Notes Parts 1-4
• Lecture Notes Parts 1-5
• Lecture Notes Parts 1-6
• Lecture notes all

Tutorials and Problems

In the tutorial, we will occasionally review topics from the lecture but mostly discuss problems and solutions to homework assignments. You are encouraged to pitch in by presenting a solution every once in a while.Every week each student 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. is required to solve homework assignments and hand them in. The problem sheets will be uploaded on Tuesdays and solutions should be turned in before the lecture on the following Tuesday. You will receive points for your solutions based on whether your solutions are correct and well-written.

Course requirements are the following: (1) You must score at least 50% of the total points of the problems assigned in each half of the semester. There will be 6 problem sheets in the first half of the semester. A sheet will have problems worth roughly 20 points. actively participate in the course. (2) You must pass an exam at the end of the semester which alone will determine your grade.

Problem Sheets

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)