# TMA4190 Introduction to Topology - Spring 2019

## 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.

## Prerequisites

You should have taken Calculus 1, 2 and 3, or something similar. 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, or compact. We will review the basics we need at the start of the course.

## Lecture Plan

This is a rough outline of the course structure that will be updated throughout the semester.

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

2.1 | Introduction | Notes 1 | ||

2.2 | Topology of Rn | Set 1 | Notes 2 | |

3.1 | Subspaces, connectedness | |||

3.2 | Cut points, compactness | Set 2 | ||

4.1 | Smooth manifolds, tangent spaces | GP 1.1 | Notes 3 | |

4.2 | Tangent spaces | GP 1.2 | Set 3 | Notes 4 |

5.1 | Inverse function theorem | GP 1.3 | Supplement | |

5.2 | Submersions | GP 1.4 | Set 4 | Notes 5 |

6.1 | Fundamental theorem of algebra | GP 1.4 | Notes 6 | |

6.2 | Transversality | GP 1.5 | Set 5 | Additional FTA notes |

7.1 | Transversality and homotopy | GP 1.6 | Notes 7 | |

7.2 | Transversality and homotopy | GP 1.6 | Set 6 | Notes 8 |

8.1 | Sard's theorem, Morse functions | GP 1.7 | ||

8.2 | Sard's theorem, Morse functions | GP 1.7 | Set 7 | Notes 9 |

9.1 | Midterm | |||

9.2 | Embeddings in Rn | GP 1.8 | Set 8,Prob 4 | Notes 9 |

10.1 | Abstract Manifolds, Projective Space | GP 1.8 | Notes 10 | |

10.2 | Partitions of Unity | GP 1.8 | Set 9 | Notes 11,P.O.U. supplement |

11.1 | Manifolds with boundary | GP 2.1 | Notes 12 | |

11.2 | Brouwer Fixed-point theorem, Transversality | GP 2.2, 2.3 | Set 10 | Notes 13 |

12.1 | Transversality | GP 2.3 | Notes 14 | |

12.2 | Mod 2 intersection theory | GP 2.4 | Set 11 | Notes 15 |

13.1 | Winding number, Borsuk-Ulam theorem | GP 2.5, 2.6 | Notes 16 | |

13.2 | Lecture cancelled | Set 12 | ||

14.1 | Orientations | GP 3.2 | Notes 17 | |

14.2 | Oriented intersection number | GP 3.3 | Set 13 | |

15.1 | Lefschetz fixed-point theory | GP 3.4 | Notes 18 | |

15.2 | Poincare-Hopf index theorem | GP 3.5 | Conway's ZIP proof |

Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Midterm, Quiz 7, Quiz 8, Quiz 9, Quiz 10, Quiz 11, Quiz 12, Course Content Outline

## Reference Group

Ling Tan

Oskar Goldhahn

Odin Hoff Gardå

## Course material

This course will follow:

- [GP] V. Guillemin and A. Pollack,
*Differential Topology*, Prentice Hall, 1974.

Other interesting and related books (not required for the course):

- [M69] J. Milnor,
*Topology from the differentiable viewpoint*, The University Press of Virginia, 1969. - [M63] J. Milnor,
*Morse Theory*, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963. - [D] B. Dundas, Differential Topology. Version January 2013
- [T] L. W. Tu,
*An Introduction to Manifolds*, Springer Verlag. - [L] J.M. Lee,
*Introduction to Smooth manifolds*, Springer-Verlag. - [S] D. Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965. - [Mu91] J.R. Munkres,
*Analysis on Manifolds*, Addison-Wesley, 1991. - [Mu75] J.R. Munkres,
*Topology: a first course*, Prentice-Hall, 1975.