Instructors: Peter Franek and Mirko Klukas, Teaching Assistant: Stephan Zhechev

The course (roughly) follows Milnor's book Topology from the differential viewpoint. Our focus is to develop an intuitive (yet solid) understanding of basic concepts in differential topology such as tangent bundles, differentials, degree of a map, and framed cobordisms.

The first goal is the Pontryagin-Thom construction which enables us to understand the space of homotopy classes of maps into spheres in some simple cases. Depending on the time-frame, other topics might be chosen in January.

**Requirements: **Regular attendence + written final exam.

Date | Topic |
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Nov. 29 | Def. of manifold, tangent space, reg. values. Proof of FTA based on counting #f^{-1}(y). |

Dec. 1 | Preimage of regular values is a submanifold. Sard's theorem and consequences: disc has FPP. |

Dec. 6 | Proof of Sard's theorem, steps 1 and 3 from Milnor (p. 16--19). Degree mod 2. |

Recitation | Classification of 1-manifolds. |

Dec. 13 | Brouwer degree: definition and basic properties. |

Recitation | sketch of classification of surfaces |

Dec. 15 | no class (we can organize a make-up class later after agreement) |

Dec. 20 | Framed cobordisms and Pontryagin-Thom construction: statement and consequences |

Recitation | Grassmanians, linking number and its properties |

Dec. 22 | Pontryagin-Thom construction: proof |

File | Comment |
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Exercises1 | Some if this will be discussed next Tuesday |