# TMA4190 Introduction to Topology - Spring 2020

Dear all,

**Updated May 25, 2020, 1:30pm**

You should all have received an email from NTNU explaining the change of format and grades for the exam. Find the same information under the Exam section.

Please note the following:

- The exam will be a digital home exam given on May 27, 9.00 - 13.00 (9am - 1pm).
- Grades have been changed from A - F to pass/fail.
- The exam will consist of a mixture of multiple choice problems and problems requiring you to show your work (see also the Exam section). Note that you will
**not**be penalized for incorrect answers for the multiple choice problems. - The syllabus will consist of chapters 1 through 8 in the lecture notes. The intended material for chapter 9 will
**not**be part of the syllabus. - Trial exam with solutions. Try working through the entire exam
**before**looking at the solutions. (There was a misprint in a previous version of the printout from Inspera Assessment for problem 4 (it should read as in the solutions with \(A = \{x\in \mathbb{Q} \mid -\sqrt 5 < x < \sqrt 5\}\) and**not**\(A = \{x\in \mathbb{Q} \mid -\sqrt 5 < x < 5\}\)). Also note that you must upload an individual file for each non-multiple choice problem. - I have also written down some solutions to the remaining problems (from chapter 7 and 8) in the lecture notes.
- I have not been able to find a satisfactory technical solution for a "digital office hour" before the exam. You are of course most welcome to send me an email about questions you might have regarding the course. Please refrain from sending me pictures with your handwriting so that your anonymity is preserved when doing the grading.

Best wishes,

Marius

Schedule | Room | ||
---|---|---|---|

Lectures: | Wednesday | 12.15 - 14.00 | EL4 |

Friday | 08.15 - 10.00 | EL4 | |

Exam: | Written exams | To be announced | |

Lecturer | |||

Marius Thaule | |||

Office: | 1248 Sentralbygg 2 | ||

Email: | marius [dot] thaule [at] ntnu [dot] no |

## What this course is about

This course is a first introduction to topology. It will start with basic concepts in point set topology (e.g. topological spaces, continuous maps, metric spaces, constructions of topological spaces, compactness, connectedness). Then it will give an introduction to algebraic topology, such as homotopy, fundamental group, and covering spaces. A short introduction to homology will give an outlook on computational methods.

See the study handbook for more information.

## What you need to know before this course

You should have seen multivariate calculus and linear algebra. Ideally you should have taken TMA4150 Algebra and/or MA3201 Rings and Modules as well, but if you haven't don't worry!

If you have any questions, just contact me!

## Lecture Plan

The first lectures will be January 8 and 10.

Lecture | Date | Topic | Notes | References |
---|---|---|---|---|

1.1 | 08.01 | Introduction, followed by an introduction to metric spaces | Chapter 1 and 2 | |

1.2 | 10.01 | From continuous maps between metric spaces to topological spaces | Chapter 2 and 3 | |

2.1 | 15.01 | Examples of topological spaces and continuous maps between topological spaces | Chapter 3 | |

2.2 | 17.01 | Continuous maps between topological spaces, homeomorphisms and closed sets | Chapter 3 | |

3.1 | 22.01 | Generating topologies | Chapter 4 | |

3.2 | 24.01 | Generating topologies. Exercises (from chapter 2) | Chapter 2 and 4 | |

4.1 | 29.01 | Exercises (from chapter 2 and 3). Constructing spaces | Chapter 2, 3 and 5 | |

4.2 | 31.01 | Constructing spaces | Chapter 5 | |

5.1 | 05.02 | Constructing spaces | Chapter 5 | |

5.2 | 07.02 | Constructing spaces | Chapter 5 | |

6.1 | 12.02 | Constructing spaces. Exercises (from chapter 4) | Chapter 4 and 5 | |

6.2 | 14.02 | Exercises (from chapter 4). Topological properties | Chapter 4 and 6 | |

7.1 | 19.02 | Postponed | ||

7.2 | 21.02 | Topological properties | Chapter 6 | |

8.1 | 26.02 | Topological properties | Chapter 6 | |

8.2 | 28.02 | Topological properties | Chapter 6 | |

9.1 | 04.03 | Topological properties | Chapter 6 | |

9.2 | 06.03 | Exercises (from chapter 5). The fundamental group | Chapter 5 and 7 | |

10.1 | 11.03 | The fundamental group | Chapter 7 | |

10.2 | 13.03 | The fundamental group | Chapter 7 | |

11.1 | 18.03 | The fundamental group | Chapter 7 | |

11.2 | 20.03 | The fundamental group | Chapter 7 | |

12.1 | 25.03 | Exercises (from chapter 6). The fundamental group of the circle | Chapter 6 and 8 | |

12.2 | 27.03 | The fundamental group of the circle | Chapter 8 | |

13.1 | 01.04 | The fundamental group of the circle | Chapter 8 | |

13.2 | 03.04 | The fundamental group of the circle | Chapter 8 | |

14.1 | 15.04 | Exercises (from chapter 7 and 8). The Seifert-van Kampen theorem* | ||

14.2 | 17.04 | The Seifert-van Kampen theorem | ||

15.1 | 22.04 | The Seifert-van Kampen theorem. Exercises |

* Not examinable.

## Course material

## References

We will not follow any particular textbook.

Some interesting books:

- [A] M.A. Armstrong,
*Basic Topology*, Springer-Verlag, 1983. - [Croo] F.H. Croom,
*Basic Concepts of Algebraic Topology*, Springer-Verlag, 1978. - [Cros] M. Crossley,
*Essential Topology*, Spring-Verlag, 2005. - [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
- [Ma] J.P. May,
*A Concise Course in Algebraic Topology*, Chicago Lectures in Mathematics, 1999.

Some books on general topology:

- [J] K. Jänich,
*Topology*, Springer, 1984. - [Mu] J.R. Munkres,
*Topology: a first course*, Prentice-Hall, 1975.

## Exam

The exam will consist of a mixture of multiple choice problems and problems that will require you to show your work. For problems that require you to show your work you can, e.g., write on a regular piece of paper and then either scan it or photograph it and upload it to your exam in Inspera Assessment. For further information about Inspera Assessment, please consult innsida.

See also here for information regarding digital home exams.

The syllabus for the exam will consist of chapters 1 through 8 in the lecture notes (including exercises).

I have prepared a trial exam for you to practice on. Try to work through it from start to finish **without** looking at the solutions.