''... homines dum docent discunt." 

                                                                                                                                                                                                                                                               Seneca    

 

                                             

Schedule of lectures:   Wednesdays  and  Thursdays  from  8:15 - 9:45  in  Gr. Hörsaal MI  (Raum 203)

 

Exercise class: Wednesdays from 10 - 11:30 in Übungsraum 2  (ehem. S3 Gyrhofstr. 8). Find the weekly homework below. 

 

Summary:  For a summary of the topics covered in each lecture, click here.

 

Office Hours: Mondays 15:00 - 17:00 

 

Teaching assistant: Dr. Thomas Rot  (contact:  thomas (dot) rot (at) uni-koeln (dot) de)

 

Language used in the lectures:  English

 

Prerequisites:  

 

 

Bibliography:

 

 For reviewing basic concepts of point-set topology:

   -  "Notes on Introductory Point-Set Topology" by Allen Hatcher, available online on his website.

   -  "Topology" by James Munkres - Pearson; 2nd edition (January 7, 2000).

 

For the course:

-  "A Basic Course in Algebraic Topology" by W.S. Massey - Graduate Texts in Mathematics,  Springer (1991).

-  "Algebraic Topology" by Allen Hatcher - Cambridge Univeristy Press (2002). Online version available.

 

 

Exams dates:   

 

• Thursday February 19th 2015, in my office 0.08, by appointment.  

 

• Monday March 23rd 2015, in my office 0.08, by appointment.

 

 

Introduction and some motivation...

 

Algebraic topology is a fundamental branch of mathematics that gives tools for understanding whether two topological spaces are ''the same'' or not.

In loose terms, to a given topological space X (think of a metric space, for instance) one can associate a list of algebraic objects (groups, rings ...), subject to the rule that if two given topological spaces are considered to be ''equivalent'' (loosely speaking, there exists a continuous deformation from one to the other), such list of associated algebraic objects should also be the same (i.e. there should be an isomorphism of the respective groups, rings ... ).

 

In this course, a particular emphasis is given to the computation of topological invariants of manifolds (e.g. real and complex projective spaces). Very often, these support finer structures, such as differentiable, symplectic, complex or Riemannian structures, the study of which goes behind the scope of this course.  However, it is worth mentioning that when two manifolds are considered to be ''equivalent" in these finer geometries (i.e. the map from one manifold to the other preserves more structure) then in particular they must be equivalent in the sense mentioned above. This makes Algebraic Topology a fundamental subject which helps the understanding of other types of geometries.

 

Also, quite surprisingly, very often the ''average" of invariants belonging to these finer geometries  only depends on weaker, topological invariants. For example, this is the case for the Gaussian curvature of a compact surface, whose average is proportional to the Euler Characteristic of the surface, and this is the content of the Gauss-Bonnet Theorem. As a consequence of it, one can prove that the torus (i.e. the exterior surface of a donut, in the picture) can never be endowed with a metric such that the Gaussian curvature is strictly positive at each point, since its

average needs to be zero.