# TMA4192 Differential Topology - Spring 2023

Tuesday 23 May: **Congrats, you did it! The exam is over!**

Here you can find the Spring 2023 Exam and the Spring 2023 Solution Suggestions.

The exam document with some feedback is here.

**Welcome to Differential Topology!**

- Last lecture:
**Friday 21 April**. - Keep checking the
**Lecture Plan**to find details of the**weekly homework**. - See the Exam Guidance section below for
**old exam papers**. - Go ahead and ask me for hints on exercises when you get stuck! This is better than going straight for the solutions.
- Feel free to either drop by my office or email me if you want to talk about anything!

**Revision session:** Tuesday 16 May

**What would you like to focus on?** Add your suggestions to the google sheet:

https://docs.google.com/spreadsheets/d/1FWq2WfRIOlaaH0u5muqPHYqoK-wEmPvglSnLOv_Iu4U/edit?usp=sharing

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

Lectures: | Thursday | 12.15-14.00 | R60 Realfag |

Friday | 12.15-14.00 | R60 Realfag | |

Exam date: | Official exam page | ||

Lecturer: | Abigail Linton | ||

Office: | 1206 in Sentralbygg 2 | ||

Email: | abigail [dot] linton [at] ntnu [dot] no |

## What this course is about

This interactive course introduces fundamental concepts and examples in differential topology and the idea of using algebraic invariants to study geometric objects. The course discusses the inverse function theorem, differentiable structures and smooth manifolds, tangent bundles, embeddings, submersions and regular/critical points. Important examples of spaces are surfaces, spheres, and projective spaces. Other key concepts are homotopy, the fundamental group and covering spaces. Applications presented in the course may range from Brouwer's fixed point theorem to vector fields on spheres. These methods and ideas have been influential to and are used in many other parts of mathematics, but also in physics and other areas of application. See the study handbook for more information.

This course is a very nice springboard for taking MA3402 Differential Forms. It also fits nicely together with the other topology courses.

## What you need to take this course

Ability to ask questions! As with many topology topics, the willingness to work to understand new terminology and ideas is almost more important than prerequisite knowledge. I expect every single student to ask at least one question each week!

Ideally you will have either taken TMA4190 Introduction to Topology already, or be taking it at the same time as this course. At the very least you should have taken Calculus 1, 2 and 3 or similar courses. For example, you should have seen what it means for a function from \( f \colon \mathbb{R}^n \to \mathbb{R}^m\) to be differentiable and what it means for a subset \(U\subseteq \mathbb{R}^n\) to be open, closed, compact, etc.

## Lecture Plan

This is a rough outline that will be updated throughout the course. Unless stated otherwise, page numbers refer to Gereon Quick's lecture notes.

Week | Topic | Pages | Homework for next Thursday | Exercises |
---|---|---|---|---|

2.1 | Introduction | 1-18 | ||

2.2 | Topology in \(\mathbb{R}^n\) and smooth maps | 19-29 | Week 2 | |

3.1 | Smooth manifolds | 30-40 | ||

3.2 | Tangent spaces and derivatives | 41-48 | Which exercise seems most interesting? Why? | 2.1, 2.2, 2.9, 2.11, 2.12 |

4.1 | Exercise session | Write up ex 2.12(b) for feedback | ||

4.2 | Differential example, Inverse Function Theorem | 48-63 | Summarise Example 3.5 to discuss | |

5.1 | Immersions and Embeddings | 63-71 | ||

5.2 | Submersions and Regular Values | 71-78 | Do exercises! | 3.2, 3.8, 3.5, 3.9, Local submersion thm proof |

6.1 | More on regular values and submanifolds | 78-83 | ||

6.2 | Exercises | Write up pf of Thm 3.24 & Exercise 3.11 | ||

7.1 | Milnor's proof of the Fundamental Theorem of Algebra | 84-90 | ||

7.2 | Transversality | 91-94 | Read p94-95. Create your own transverse example to discuss | |

8.1 | Transversality of submanifolds | 94-98 | ||

8.2 | Examples & The Theorem of Sard and Brown | 98-105 | 3.17, 3.18, 3.20, 3.19 | |

9.1 | Smooth Homotopies | 114-123 | ||

9.2 | Stable properties. Tangent bundles | -124, 53-58 | 4.1, 4.3, 4.5 | |

10.1 | Exercises, Abstract Manifolds | 127-130 | Write up Exercise 3.17(c ) | |

10.2 | Real Projective Space, Stiefel Manifolds | 130-137 | ||

11.1 | Stiefel Manifolds, Grassmannians | 137-145 | Read this Quanta article | |

11.2 | Partition of unity, Whitney Embedding | 145-153 | Compare proofs Thm 5.30 and Thm 5.26 | 5.1, 5.2 |

12.1 | Manifolds with Boundary | 161-167 | ||

12.2 | Regular values on manifolds with boundary | 167-171 | 6.5, 6.1, 6.2 | |

13.1 | Brouwer Fixed Point Theorem | 171-178 | ||

13.2 | Exercises | Fibres of Hopf fibration | ||

14.1 | Easter break | |||

14.2 | Easter break | |||

15.1 | Intersection Numbers modulo 2 | 241-246 | ||

15.2 | Intersection Numbers mod 2 examples | 245-247* | *Note: this lecture covered examples not in the notes! | 9.1, 9.3, 9.4, 9.5(b) |

16.1 | Degrees modulo 2, Exercises | 186-191 | ||

16.2 | Revision Exercises | |||

Extra | Example exam question |

## Reference Group

We will have three reference group meetings over the semester. Either talk to Abi or one of the reference group members if you have anything you would like brought up such as what you think about the pace, course content and exercises, or fun ideas for activities!

**Members:**Sunniva Engan, Bjørnar Ørjansen Kaarevik, Inga Maria Tillmann**First meeting date:**Week 6**Second meeting date:**Week 10**Third meeting date:**Week 17

## Exam Guidance

**Update 23 May 2023**: Here you can find the Spring 2023 Exam and the Spring 2023 Solution Suggestions

**Update 2 June 2023**: The exam document with some feedback is here.

**Exam format:** You should attempt all questions for a total of 100 marks. An example question is here: Example Exam Question

The exercises, examples, counter-examples and proofs from lectures are great preparation: you will definitely be seeing some of them again on the exam! Their style and the way we have talked about examples/proofs etc in class is very representative of the style of the questions on the exam.

There are a number of **relevant old exam papers** from an old version of this course.
Note that the first question on the early papers are now covered by the Intro to Topology course, and some questions e.g. on the 2010, 2011 and 2014 papers are not within our scope.

Feel free to write up an example/exercise/proof and submit/show it to me to **get feedback** on how written solutions should look in the exam. (Disclaimer: I make no guarantees about how quickly I can get it back to you, especially in the lead-up to the exam!)

## Course material

We will mostly be following Gereon Quick's lecture notes from a previous course. It is usually beneficial to compare different sources and find the style that most suits you. This course is mainly influenced by these books:

- [GP] V. Guillemin and A. Pollack,
*Differential Topology*, Prentice Hall, 1974. - [M] J. Milnor,
*Topology from the differentiable viewpoint*, The University Press of Virginia, 1969. - [T] L. W. Tu,
*An Introduction to Manifolds*, Springer Verlag.

Some other interesting books are:

- [BM] J. Baez, J. Muniain,
*Gauge fields, knots and gravity*, World Scientific Publishing, 1994. - [D] B. Dundas,
*Differential Topology*, Version 2017. - [MM] J. Milnor,
*Morse Theory*, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963. - [BJ] T. Bröcker, K.Jänich,
*Introduction to differential topology*, Cambridge Univ. Press, 1982. - [L] J.M. Lee,
*Introduction to Smooth manifolds*, Springer-Verlag. - [S] D. Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965.

Some books on general topology:

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

For a bit of a topology-related break:

- Jeff Weeks's geometry games.