## Differential Topology [MA5122]

### Sommersemester 2014

### Prof. Dr. Michael M. Wolf

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

Übungsleitung: |
Alexander Müller-Hermes | |

Mitwirkende: |
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Vorlesung: |
Mittwoch, 10:15 - 11:45, Raum 00.07.011 | Anmeldung |

Zentralübung: |
Montag, 14:15 - 15:45, Raum 03.08.011 | Anmeldung |

Zweite Zentralübung: |
Freitag, 08:30-10:00, Raum 03.10.011 |

### News:

- The repeat (oral) examination will be on September 25.

You can register in TUMonline from September 1 - 15.

- Until 7.7. there is a doodle poll for date of the examination (which, if it is oral, will have to be in the week around July 15th).

If you have signed up for the examination, you should have received an email with the link to the poll.

### 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.File | Version | Topics |
---|---|---|

lecture 1 | 09.04.2014 | Introductory remarks, reminder of topological spaces, topological manifolds |

lecture 2 | 16.04.2014 | quotient topology, embedding of topological manifolds, differential calculus |

lecture 3 | 30.04.2014 | manifolds with boundary, differentiable structures, smooths manifolds and smooth maps |

lecture 4 | 07.05.2014 | smooth invariance of domain and its consequences, embeddings, immersions, submersions |

lecture 5 | 14.05.2014 | smooth submanifolds |

lecture 6 | 21.05.2014 | tangent space, tangent bundle, differential |

lecture 7 | 28.05.2014 | Sard's theorem, preimage theorem with boundary, no-retraction theorem |

lecture 8 | 04.06.2014 | Brouwer's fixed point theorem, Whitney's embedding theorem I |

lecture 9 | 11.06.2014 | Whitney's embedding theorem II, homotopy, isotopy, mod-2 degree |

lecture 10 | 25.06.2014 | mod-2 winding number, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem |

lecture 11 | 02.07.2014 | Corollaries from Borsuk-Ulam, Ham-Sandwich theorem, orientability of manifolds |

lecture 12 | 09.07.2014 | Brouwer degree, fixed points on S^n, hairy ball theorem, Hopf's characterization of homotopy classes into S^n |

Exercises | Solutions | Due date | Topics |
---|---|---|---|

Exercise 1 | Solution 1 | 23.04.2014 | Topology, real projective Space, matrix groups, a riddle |

Exercise 2 | Solution 2 | 07.05.2014 | Partition of unity, bump functions, differential calculus, incompatible differential structures |

Exercise 3 | Solution 3 | 21.05.2014 | Classification of smooth and compact 1-manifolds, projective spaces, manifolds with boundary, immersions and embeddings |

Exercise 4 | Solution 4 | 04.06.2014 | Lie groups, Lie group actions, embedding of RP4, maxima and minima, Grassmann manifold |

Exercise 5 | Solution 5 | 18.06.2014 | Transversality, Stack of records theorem, Morse functions, Brouwer |

Exercise 6 | Tutor 6 Homework 6 | 02.07.2014 | Embedding problems, rectangle on curve, Homotopy, smooth-no-retraction, mod 2 fundamental theorem of algebra |

Exercise 7 | Tutor 7 Homework 7 | 09.07.2014 | Ham-Sandwich, Orientations, Degree of a map |

### Literature:

- Notes

- 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