About the Seminar

The basics of topology (often referred to as point set topology) have the merit that they can be usefully applied in a wide variety of contexts. On the other hand true topology offers much deeper geometric insights, and almost invariably means algebraic or differential topology -- indeed often a combination of the two. While the former is regularly taught at our department the latter is not, and the seminar will be a unique chance to learn what differential topology is about.

The aim of the seminar, therefore, is modest: participants shall become familiar with the basic ideas and most common tools of differential topology. Thus no sources beyond textbooks will be required, and indeed the seminar will, by and large, be organized along the lines of the book by Bröcker and Jänich. Subjects that will be treated include differential manifolds, vector bundles, construction of diffeomorphisms, connected sums and surgery.

Participants must have a solid knowledge of real analysis (several variables) and linear algebra, as well as point set topology (e.g. covered by the ''Einführung in die Topologie'' course). Beyond that, familiarity with basic notions of ODEs may be useful but will not be assumed. No algebraic topology will be used; the seminar rather provides a complementary view of topology, from quite a different (and appreciably more geometric) angle.

Preliminary List of Talks

1. Nadine Cremer: Manifolds & Differentiable Structures [BJ1]
   
   
2. Oleksandr M1: The Tangent Space [BJ2]
   
   
3. Sascha Feth / Michael Kerber: Vector Bundles [BJ3]
   
   
4. Oleksandr Iena / Oleksandr M2: Linear Algebra of Vector Bundles [BJ4]
   
   
5. L. Ensthaler: Local and Tangential Properties [BJ5]
   
   
6. Marc Dohm: Sard's Theorem [BJ6]
   
   
7. Thorsten Horberth: Embedding [BJ7]
   
   
8. Frank Seifried: Dynamical Systems [BJ8]
   
   
9. Valentin Tonita: Isotopies of Embeddings [BJ9]
   
   
10. Alicia Nieto: Connected Sum [BJ10]
    
    
11. Eva-Maria Zimmermann: Classification of Compact Surfaces

Contents of the individual talks may still be adjusted, and in any case participants involved in the subjects 3, 4, 7, or 11 should consult me about this in good time.

Quite generally I expect every speaker to show and discuss with me a rough plan of the proposed talk not later than four weeks before the scheduled date, and a detailed draft two weeks later. (Of course you are welcome to see me at any time about whatever you want to discuss.)

Every speaker is requested to complement his or her talk by posing one or two illuminating but easy problems to the audience. Participants are expected to hand in written solutions within the week: the speaker will have fun to mark their work.

Literature

• Th. Bröcker and K. Jänich: Einführung in die Differentialtopologie. (A copy of the English edition has been ordered.)
  
  
• S. Lang: Differential Manifolds.
  
  
• M.W. Hirsch: Differential Topology.

Summary

The aim of this course is to introduce students to standard methods in topology, with a view towards both further studies in that field and applications in other, related branches of mathematics including singularity theory and algebraic geometry.

True topology is often called algebraic because it is based on the surprising fact that topology, an inherently geometric concept, is governed by rich algebraic structures. Under the name of homology these structures can be made quite explicit, and the vast majority of results in topology, and topological results in other fields, depend in some way on calculations of homology. Euler's famous theorem on polyhedra is a first example: if v, e, and f respectively denote the number of vertices, edges, and faces of a compact polyhedron then v-e+f=2.

Homology will therefore be the central topic of the course. Among numerous possibilities to explain homology, none very simple but some very complicated, I have opted for the simplest and most explicit one, which is based on the notion of cell complexes.

Students who choose this course must be thoroughly familiar with undergraduate analysis and linear algebra. They should also have a good knowlegde of basic topological concepts commonly subsumed under the name of general or point set topology, including neighbourhoods, separation of points, connectedness, compactness, and quotient topological spaces. All these notions are covered by the two hour course "Einführung in die Topologie" I have taught the previous term, and in fact (and not surprisingly) that course essentially defines the prerequisites for the one to come. Thus you may want to check details by consulting the corresponding lecture notes (see below). These notes were specifically written with German and international students in mind who wish to join the topology course at the beginning of the winter term. The text is self-contained and if you do not, or not fully meet the requirements as stated you may find the notes suitable for preparatory self-study. I will also be happy to give further information and personal assistance, and you should not hesitate to contact me at any time if you have any doubts or queries.

Lecture Notes

A complete set of notes is provided for students' use, in two parts:

• Homotopy
• Homology

Similar notes (also in English) of my previous course "Einführung in die Topologie" are available too:

• A Topology Primer

Problem Sheets

• Einführung in die Topologie (Winter 2001/2002)
• Topology (Summer 2002)

Timetable

Literature

• M.A. Armstrong: Basic Topology. Springer 1997
• N. Bourbaki: Topologie Générale, chapitres 1 - 4. Hermann 1971
• Th. Bröcker and K. Jänich: Einführung in die Differentialtopologie. Springer 1990
• T. tom Dieck: Topologie. De Gruyter1991
• T. tom Dieck, Klaus Heiner Kamps und Dieter Puppe: Homotopietheorie. Springer 1970
• A. Dold: Lectures on Algebraic Topology. Springer 1972
• A. Fomenko: Visual Geometry and Topology. Springer 1994
• M.W. Hirsch: Differential Topology. Springer 1997
• K. Jänich: Topology. English edition out of print
• J.L. Kelley: General Topology. Springer 1975
• W.S. Massey: Algebraic Topology: An Introduction. Harcourt, Brace & World 1967
• H. Schubert: Topologie. Teubner 1975
• R.M. Switzer: Algebraic Topology - Homotopy and Homology. Springer 1975, reprinted 2002