Discrete Geometry Group

BMS Basic Course “Topology”
Winter Term 2008/2009

Times and places

Carsten Schultz: Mon 8-10 & 10-12; MA 841
The last lecture will take place on Mon, Feb 9, 10-12. (The written exam will take place from 8 to 10, see below.)

Tutorials by Axel Werner: Wed 12-14, MA 651; Wed 14-16, MA 644.

Exams

The TU Bachelor students will have oral exams as a ‘Modulprüfung’.

Everyone else will take the written exam on Monday, Feb 9, 8ct. The written exam will take between 60 and 80 minutes.

Of course, for both groups of students it will be necessary to have scored enough points on the homework problems. If you are in doubt, ask Axel.

Overview

We first give an introduction to general topology, also known as point set topology. This is the theory of topological spaces. Concepts like continuity or compactness are most naturally treated in this context. We revise these and then investigate several ways to construct new spaces out of given ones. We can glue spaces together, for example we glue together simplices to obtain simplicial complexes, a combinatorial way of describing spaces. We can specify a space by prescribing which maps into it are continuous, or which sequences in it should converge. This is a technique often used in functional analysis (e.g. `topologies of point-wise convergence').

We then take first steps in algebraic topology, which studies spaces via associated algebraic structures. We get to know the fundamental group of a space, which is already essential in the study of surfaces. We also define the homology groups of a simplicial complex, which will enable us to prove theorems on the non-existence of certain maps, for example Brouwer's fixed point theorem.

Lecture Notes

Complete lecture notes, one big PDF file.

These were prepared during the course of the semester. They are based on the notes for an earlier class at FU Berlin and hence be in German.

I thank everyone who suggested corrections and improvements, in particular Axel, Florian, and Thomas.

Problem sets

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References

A short bibliography can be found at the end of part 1 of the lecture notes.