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Algebraic Topology I, WS 2015/16

Prof. Dr. C. Löh

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Algebraic Topology

Algebraic topology studies topological spaces via algebraic invariants -- by modelling certain aspects of topological spaces in the realm of algebra, e.g., by groups and group homomorphisms. Classical examples include homotopy groups and (co)homology theories.

Moebius

Algebraic topology has various applications, both in theoretical and in applied mathematics, for instance, through fixed point theorems and (non-)embeddability results. For example, Nash's proof of existence of certain equilibria in game theory is based on a topological argument. Topics covered in this course include: This course will be continued in the summer 2016 with the course "Algebraic Topology II".

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Summary (in German): pdf. Topics covered so far:

Time/Location

Tuesday, 10--12, M 104,
Friday, 10--12, M 104.

Exercise classes

Wed, 16--18, M102. First exercise class on Oct 14.

The exercise class will be held by Dr. Dmitri Pavlov and Gerrit Herrmann.

Exercise sheets

Solutions can be submitted in English or German and in teams of up to two people. Please do not forget to add your name to all your submissions!

Sheet -1, of October 13, 2015, no submission, will be discussed in the exercise class on October 14.
Sheet 0, of October 16, 2015, no submission, will be discussed in the exercise class on October 21.
Sheet 1, of October 16, 2015, submission by October 23, will be discussed in the exercise class on October 28.
Sheet 2, of October 23, 2015, submission by October 30, will be discussed in the exercise class on November 4.
Sheet 3, of October 30, 2015, submission by November 6, will be discussed in the exercise class on November 11.
Sheet 4, of November 6, 2015, submission by November 13, will be discussed in the exercise class on November 18.
Sheet 5, of November 13, 2015, submission by November 20, will be discussed in the exercise class on November 25.
Sheet 6, of November 20, 2015, submission by November 27, will be discussed in the exercise class on December 2.
Sheet 7, of November 27, 2015, submission by December 4, will be discussed in the exercise class on December 9.
Sheet 8, of December 4, 2015, submission by December 11, will be discussed in the exercise class on December 16.
Sheet 9, of December 11, 2015, submission by December 18, will be discussed in the exercise class on January 13.
Sheet 10, of December 18, 2015, submission by January 8, will be discussed in the exercise class on January 13.
Sheet 11, of January 8, 2016, submission by January 15, will be discussed in the exercise class on January 20.
Sheet 12, of January 15, 2016, submission by January 22, will be discussed in the exercise class on January 27.
Sheet 13, of January 22, 2016, submission by January 29, will be discussed in the exercise class on February 3.
Sheet 14, of January 29, 2016, no submission
Sheet 15, of February 5, 2016, no submission

Literature

This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books.

Algebraic Topology Point-set Topology Homological Algebra Category Theory

Prerequisites

All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Knowledge about manifolds as in Analysis IV is not necessary, but helpful.

Exams

Please read the information on organisation and formalities of this course.

Dates for the oral exams (25 minutes): Registration details: . If you prefer to take a combined oral exam on Algebraic Topology I/II, please contact me.

Last change, February 5, 2016

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