''... 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. ## S. Sabatini

(contact: sabatini (at) math (dot) uni-koeln (dot) de)

(Wintersemester 2014/2015)

Oral Exams: in my office 0.08, see below!

Homework: In every homework assignment you will find "Mandatory" and "Optional" exercises (- if no specification is made it means that all exercises are mandatory -).

"Mandatory" means that you should hand in the exercise, it will be corrected and it counts in the final homework grade.

"Optional" means that you should do it to practice more, it will be corrected but it will *not* count in the final homework grade.

Homework 0 (No due date)

Homework 1(Due on October 22nd in class)

Homework 2 (Due on October 29th in class).

Errata corrige: In Exercise 5 I actually meant "path-connected" and not "arc-connected".

Homework 3 (Due on November 5th in class).

Homework 4 (Due on November 12th in class)

Homework 5 (Due on November 19th in class)

Homework 6 (Due on November 26th in class)

Homework 7 (Due on December 3rd in class)

Homework 8 (Due on December 10th in class)

Homework 9 (Due on December 17th in class)

Homework 10 (Due on January 7th in class)

Homework 11 (Due on January 14th in class)

Homework 12 (Due on January 21st in class)

Homework 13 (Due on January 28th in class)

Homework 14 (Due on Friday February 6th at noon in Mrs. Angela Georg's office "Geschaeftszimmer" in Prof. Sabatini's mail box. Graded by Monday February 9th. Make an appointment with Dr. Rot to have the homework back on Monday 9th.)

A good knowledge of Point-set Topology is strongly recommended - for example the notions studied in the analysis courses (although some basic definitions and facts will be recalled in class), as well as basic algebraic concepts (e.g. notions of group, ring, module) should be familiar to the student.

During the first week of classes we will see (without proof) another manifestation

of the same phenomenon: the Poincaré-Hopf Theorem, a consequence of it being

the so called ''Hairy ball theorem": every nonzero vector field on a two dimensional

sphere has at least one zero (whereas on the torus one can have infinitely many

nowhere vanishing vector fields).

The study of similar concepts - with a differential point of view - will be treated in

"Torus". Licensed under Public domain via Wikimedia Commons -

http://commons.wikimedia.org/wiki/File:Torus.png#mediaviewer/File:Torus.png