WEEK |
BRIEF DESCRIPTION |
notes/homeworks/solutions |

before starting | Refreshing topology (based on Munkres: Topology 2nd Ed.) Chap 41: locally-finite, paracompactness, normal/regular spaces; Chap 36: imbeddings of manifolds, partition of unity; Chap 50: dimension theory | |

01/12- 01/18 | partition of unity | |

01/19- 01/25 | 1.1 #3, 4, 11, 12; 1.2 # 4, 5 | |

01/26- 02/1 | Submersion theorem, preimage theorem, cutting out submanifolds | 1.2 #11; 1.3 # 2, 5, 10; 1.4 #2, 6, 7 |

02/2- 02/8 | Transversality, stability and homotopy | |

02/9- 02/15 | Sard's theorem, Whitney embedding theorem (Umich math 437W13 note, UToronto math 1300 note, an explicit construction of injective immersion is missing from Victor's book, should read these notes), Transversality (section 2.3) | 1.5 #6, 7, 9, 10; 1.6 #3, 7, 8 |

02/16- 02/22 | ||

02/23- 03/01 | transversality and Sard's for manifolds-with-boundary; Classification of compact one-manifolds, Brouwer fixed point theorem | 2.1 #2, 8, 10, 11; 2.2 #2, 3, 6, 7 |

03/02- 03/08 | The transversality theorem | 2.3 #2,3,5,12,18,20 |

03/09- 03/15 | Intersection theory Mod 2, Jordan-Brouwer separation theorem |

Links to similar courses:

Geometry of manifold (MIT) U of Toronto U of Toronto handouts of some basic concepts in differential geometry

List of some basic problems:

dimension of tangent space is the same as the manifold