# TMA4192 Differential Topology - Spring 2022

**Welcome to Differential Topology!**

**Classes start on Thursday, January 13**. The first two weeks (at least) will be**only online**on zoom. Further info will follow soon, both here and via Blackboard. So do sign up and check this page regularly! :)- The course will follow these Lecture Notes. However, they will be updated regularly during the course. The notes are mainly based on the book by Guillemin and Pollack [GP].
- You can use this course to fulfil the requirements for the Master degree to take classes in Geometry & Topology. For more details contact Gereon in his capacity as the academic study adviser for the Master program.
- Please make sure you chose your spot in the exam schedule and contact Gereon as soon as possible if there are any problems or questions.

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

Lectures: | Thursday | 12.15-14.00 | zoom and H3 Hovedbygg |

Friday | 12.15-14.00 | zoom and R4 Realfagsbygg | |

Office Hours: | upon request | ||

Exam: | oral exams | schedule | |

Lecturers | |||

Gereon Quick | Melvin Vaupel | ||

Office: | 1246 in Sentralbygg 2 | ||

Email: | gereon [dot] quick [at] ntnu [dot] no | melvin [dot] vaupel [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 tentative lecture plan. We will most likely be a bit slower than scheduled. Here are the collected Lecture Notes.

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

2.1 | Introduction | Lecture 01 | ||

2.2 | Topology in \(\mathbb{R}^n\) and smooth maps | GP 1.1 | Lecture 02 | |

3.1 | Smooth manifolds | GP 1.1 | Set 01 + Soln | Lecture 02 + Lecture 03 + Notes 03 |

3.2 | Examples of smooth manifolds | GP 1.1 | Lecture 03 + Set 01 | |

4.1 | Tangent spaces and derivatives | GP 1.2 | Set 02 + Soln | Lecture 04 |

4.2 | The Inverse Function Theorem and Immersions | GP 1.3 | Lecture 05 | |

5.1 | Immersions and Embeddings | GP 1.3 | Set 03 + Soln | Lecture 06 |

5.2 | Submersions and Regular Values | GP 1.4 | Set 04 + Soln | Lecture 07 |

6.1 | More on regular values and submanifolds | GP 1.4 | Lecture 07 + Lecture 08 | |

6.2 | Lie groups | Set 05 + Soln | Lecture 08 | |

7.1 | Milnor's proof of the Fundamental Theorem of Algebra | M 1.3 + GP 1.4 | Lecture 09 | |

7.2 | Transversality | GP 1.5 | Set 06 + Soln | Lecture 10 |

8.1 | Transversality of submanifolds | GP 1.5 | Lecture 11 | |

8.2 | The Theorem of Sard and Brown | GP 1.7 | Lecture 12 | |

9.1 | Smooth Homotopies | GP 1.6 | Set 07 + Soln | Lecture 13 |

9.2 | Tangent Bundles and Embedding Manifolds in Euclidean Space | GP 1.8 | Lecture 14 | |

10.1 | Abstract Manifolds and Real Projective Space | Set 08 + Soln | Lecture 15 | |

10.2 | Stiefel Manifolds and Grassmannians | Lecture 16 | ||

11.1 | Partition of unity and Whitney's Embedding Theorem | GP 1.8 | Lecture 17 | |

11.2 | Manifolds with Boundary: Motivation | GP 2.1 | Set 09 + Soln | Lecture 18 |

12.1 | Manifolds with Boundary and one-manifolds | GP 2.2 + M 2 | Lecture 18 + Lecture 19 | |

12.2 | Brouwer Fixed Point Theorem | GP 2.2 + M 2 | Set 10 + Soln | Lecture 19 |

13.1 | Intersection Numbers modulo 2 | GP 2.4 + M 4 | Set 11 + Soln | Lecture 21 |

13.2 | Degrees modulo 2 | GP 2.4 + M 4 | Lecture 21 | |

14.1 | Intersection Numbers: Examples and Exercises | GP 2.4 + M 4 | Lecture 21 + Set 11 | |

14.2 | Winding Numbers and the Borsuk-Ulam Theorem | GP 2.5 + 2.6 | Set 12 + Soln | Lecture 22 |

15.1 | Easter Break | |||

15.2 | Easter Break | |||

16.1 | Outlook: Orientations | GP 3.2 + M 5 | ||

16.2 | Outlook: Intersection Theory revisited | GP 3.3 + M 5 |

## Reference Group

- Members: Naglius Kirvaitis Brandal, Eivind Xu Djurhuus, Mikkel Elkjær Kårstein

- 1st reference group meeting: 09 February 2022 at 13.15
- 2nd reference group meeting:
- 3rd reference group meeting:

## 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:

- [BM] J. Baez, J. Muniain,
*Gauge fields, knots and gravity*, World Scientific Publishing, 1994. - [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. - [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.