## Differential Topology [MA5122]

### Wintersemester 2018/19

### Prof. Dr. Michael M. Wolf

Lecturer: |
Prof. Dr. Michael M. Wolf | |

Assistant: |
Margret Heinze | |

Lecture: |
Mo 14:15 - 15:45 (room 00.09.022) | Anmeldung |

Exercises: |
Mi 10:15-11:45 und Do 08:15-09:45 (room 03.12.020A) | Anmeldung |

### News

- Exam will be written on March 1st, 10:00 - 11:00 in room 00.04.011 (MI lecture hall 2). Notes, books, electronic devices, etc. will not be allowed.

### Content

We will follow a direct approach to differential topology and to many of its applications without requiring and exploiting the abstract machinery of algebraic topology. The course will cover immersion, submersions and embeddings of manifolds in Euclidean space (including the basic results by Sard and Whitney), a discussion of the Euler number and winding numbers, fixed point theorems, the Borsuk-Ulam theorem and respective applications. At the end of the course, students should be able to analyse topological problems from a differentiable viewpoint and to see differential problems from a topological perspective.Date | Content | Black board notes | Further reading |
---|---|---|---|

Oct 15 | Intro, Topological spaces, subspace-, product- and quotient-topologies | lec 1 | A. Hatcher's notes ^{} with more details and proofs |

Oct 22 | Compactness, homeomorphisms, topological manifolds and their embeddings into Rn, differential calculus | lec 2 | |

Oct 29 | Classification of low-dim. manifolds, manifolds with boundary, constant rank thm., smooth structures | lec 3 | |

Nov 5 | Smooth maps and diffeomorphisms, smooth invariance of domain and its consequences | lec 4 | |

Nov 12 | Embeddings, immerions, submersions, submanifolds, preimage theorems | lec 5 | |

Nov 19 | Tangent-vectors, -spaces and -bundles | lec 6 | |

Nov 26 | Differentials, measure zero sets, Sard's theorem with and without boundary | lec 7 | |

Dec 3 | Density of Morse functions, Morse's Lemma, no-retraction theorem, Brouwer's fixed point theorem | lec 8 | |

Dec 10 | Invariance of domain and dimension, Whitney's embedding and immersion theorem | lec 9 | |

Jan 7 | Smooth homotopy and isotopy, mod2-degree | lec 10 | |

Jan 14 | Mod-2 winding numbers, Jordan-Brouwer separation, Borsuk-Ulam 1st version | lec 11 | |

Jan 24 | Borsuk-Ulam variants, optimal non-linear approximations, parametrized Sard for probability-one homotopy methods | lec 12 | |

Jan 28 | Ham-sandwich theorem, Orientability, Brouwer degree theorem | lec 13 | |

Feb 4 | Properties of Brouwer degree, hedgehog theorem, Euclidean degree, boundary theorem, Rothe's theorem | lec 14 | Chap.1 in Vandervorst's online book ^{} |

### Exercises

- Additional exercise sessions on Feb 6th 10:15-11:45 and Feb 7th 08:30-10:00 in room 03.12.020A
- Exercise sessions will be held every two weeks starting in the week from 22nd Oct - 26th Oct
- Date, time and room are: Wed 10:15-11:45 in room 03.12.020A and Thu 08:30-10:00 in room 03.12.020A
- Exercise sheets are to prepared at home and will be discussed during exercise sessions

File | Content | Week | Solution |
---|---|---|---|

Ex 1 | Basic notions of topologies | Oct. 22nd - Oct. 26th | Sol 1 |

Ex 2 | Topological and smooth manifolds | Nov. 5th - Nov. 9th | Sol 2 |

Ex 3 | Lie groups, embeddings, immersions, submanifolds | Nov. 19th - Nov. 23th | Sol 3 |

Ex 4 | tangent space, more Lie groups | Dec. 3th - Dec. 7th | Sol 4 |

Ex 5 | Brouwer's fixed point theorem, classifiaction of 1-dim compact manifolds | Dec. 17th - Dec. 21th | Sol 5 |

Ex 6 | embeddings, Nash equilibrium | Jan. 16th - Jan. 17th | Sol 6 |

Ex 7 | stack of records, smooth homotopy, mod-2 degree | Jan. 28th - Feb. 1st | Sol 7 |

Ex 8 | degree, Borsuk-Ulam | Feb. 4th - Feb. 8th | Sol 8 |

### Additional literature

- M.W. Hirsch, Differential Topology
- V. Guillemin, A. Pollack, Differential Topology
- J.W. Milnor, Topology from the differentiable viewpoint; video recordings of a classic lecture by Milnor can be found here: part I
^{}, part II^{}, part III^{} - Bröcker, Jänich, Einführung in die Differentialtopologie
- Munkres, Topology
- Deimling, Non-linear Functional Analysis (for Euclidean degree)