# TMA4192 Differential Topology - Spring 2021

**Welcome to Differential Topology!**

- This is the place where you will find all messages concerning the course.
- Here is an
**updated version**of the Lecture Notes. The course will follow these notes which are mainly based on the book by Guillemin and Pollack [GP]. - The
**exam**form has been changed from written to**oral exam**in order to have a safe and predictable exam. If you don't like oral exams, don't worry, it'll be OK. :) - Delta students created a
**Maths Discord**. It has a channel for our class as well. Here are the coordinates to sign up. - For the time being, the lectures will take place on campus as initially scheduled (Friday starts at 08.30) in the timetable. The lectures will also be streamed via zoom as usual. Please remember to
**use the Check-in**options when you are on campus! - Here is a suggestion for a schedule for the
**final exam**. Please fill in your full name in preferred time slot as soon as possible. Please also contact me if you have any question, time conflict, etc.

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

Lectures: | Monday | 12.15-14.00 | online via zoom and in B1 Berg |

Friday | 08.30-10.00 | online via zoom and in MA24 Grønnbygget | |

Office Hour: | upon request | online via zoom | |

Exam: | oral exams | ||

Lecturer | |||

Gereon Quick | |||

Office: | 1246 in Sentralbygg 2 | ||

Email: | gereon [dot] quick [at] ntnu [dot] no |

## What this course is about

The aim of the course is to introduce 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.

## What you need to know before this course

Ideally, you will have taken TMA4190 Introduction to Topology before or at the same time with this class. But I will not assume that you have done so. :) Hence my goal is to assume as little previous knowledge as possible, but 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,… :)

## Lecture Plan

This is our lecture plan. Note that we deviated slightly at the end. Here are the collected Lecture Notes.

Week | Topic | Reference | Exercises | Notes |
---|---|---|---|---|

2.1 | Introduction | Lecture 1 | ||

2.2 | Topology in \(\mathbb{R}^n\) and smooth maps | GP 1.1 | Lecture 2 | |

3.1 | Smooth manifolds | GP 1.1 | Set 1 + Soln | Lecture 3 |

3.2 | Lecture is cancelled | |||

4.1 | Tangent spaces and derivatives | GP 1.2 | Set 2 + Soln | Lecture 4 |

4.2 | The Inverse Function Theorem and Immersions | GP 1.3 | Lecture 5 | |

5.1 | Immersions and Embeddings | GP 1.3 | Set 3 + Soln | Lecture 6 |

5.2 | Submersions and Regular Values | GP 1.4 | Lecture 7 | |

6.1 | Milnor's proof of the Fundamental Theorem of Algebra | M 1.3 + GP 1.4 | Set 4 + Soln | Lecture 8 |

6.2 | Transversality | GP 1.5 | Lecture 9 | |

7.1 | Transversality of submanifolds | GP 1.5 | Set 5 + Soln | Lecture 10 |

7.2 | Homotopy and Stability | GP 1.6 | Set 6 + Soln | Lecture 11 |

8.1 | Sard's Theorem and Morse functions | GP 1.7 | Lecture 12 | |

8.2 | Tangent Bundles and Embedding Manifolds in Euclidean Space | GP 1.8 | Lecture 13 | |

9.1 | Abstract Manifolds and Real Projective Space | Set 7 + Soln | Lecture 14 | |

9.2 | Stiefel Manifolds and Grassmannians | Lecture 15 | ||

10.1 | Partition of unity and Whitney's Embedding Theorem | GP 1.8 | Lecture 16 | |

10.2 | Manifolds with Boundary | GP 2.1 | Set 8 + Soln | Lecture 17 |

11.1 | Brouwer Fixed Point Theorem and One-Manifolds | GP 2.2 + M 2 | Lecture 18 | |

11.2 | Transversality is generic | GP 2.3 | Set 9 + Soln | Lecture 19 |

12.1 | Intersection Numbers and Degrees modulo 2 | GP 2.4 + M 4 | Set 10 + Soln | Lecture 20 |

12.2 | Intersection Numbers: Examples and Exercises | GP 2.4 + M 4 | L20 and Set 10 | |

13.1 | Easter Break | |||

13.2 | Easter Break | |||

14.1 | Easter Break | |||

14.2 | Winding Numbers and the Borsuk-Ulam Theorem | GP 2.5 + 2.6 | Set 11 + Soln | Lecture 21 |

15.1 | Orientations | GP 3.2 + M 5 | Set 12 + Soln | Lecture 22 |

15.2 | Intersection Theory, Euler characteristic and Degrees | GP 3.3 + M 5 | Set 13 + Soln | Lecture 23 |

16.1 | Lefschetz Fixed-Point Theory | GP 3.4 | Lecture 24 | |

16.2 | Vector bundles and the Poincare-Hopf Theorem | GP 3.5 | Lecture 25 | |

17.1 | Hopf Degree Theorem | GP 3.6 + M 7 | Lecture 26 |

## Reference Group

Johanna Aigner, Leona Rodenkirchen, Robin Fissum

- 1st reference group meeting: Friday, 5 February 2021, at 10.00.
- 2nd reference group meeting: Friday, 19 March 2021, at 10.00.
- 3rd reference group meeting: Monday, 7 June 2021, at 10.00

## Course material

Main sources:

- [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.

A highly recommended companion to these books is:

- [T] L. W. Tu,
*An Introduction to Manifolds*, Springer Verlag.

Some other interesting books are:

- [D] B. Dundas, Differential Topology. Version January 2013
- [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. - [G] R. Ghrist,
*Elementary Applied Topology*. - [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.

Jeff Weeks's geometry games.