## MAT 566: Differential Topology
## Stony Brook Fall 2006 |
---|

Problem Set 7 is due on Thursday, December 14, by 5pm, in Math 3-117.

Here are some Notes on the Splitting Principle.

Here is general information about the course
(*revised 9/7*).

All readings listed in the table below are from Milnor&Stasheff's
*Characteristic Classes*, except

- Munkres: Munkres,
*Elements of Algebraic Topology* - G&H: Griffiths&Harris,
*Principles of Algebraic Geometry* - Vakil's slides: Vakil,
*slides for 2005 MAA Inivited Address* - Spanier: Spanier,
*Algebraic Topology* - Expository Notes: Zinger,
*Counting Plane Rational Curves* - Atiyah-Bott: Atiyah&Bott,
*The Moment Map and Equivariant Cohomology*, Topology 1984 - Mirror Symmetry: K. Hori, et. al.,
*Mirror Symmetry* - Milnor's paper: Milnor,
*On manifolds homeomorphic to the 7-sphere*, Annals of Math, 1956 - Warner: Warner,
*Foundations of Differentiable Manifolds and Lie Groups* - Spin Geometry: Lawson&Michelsohn,
*Spin Geometry*

*Name:*
Aleksey Zinger
*E-mail:* azinger@math
*Phone:* 432-8618

*Office:* Math Tower 3-117
*Office Hours:* Wed 9-12

*Problem Set 7 is due on Thursday, December 14, by 5pm, in Math 3-117*

Date |
Topic |
Read |
Problem Set |

9/7 | Review of Vector Bundles | Sections 1-3 | pdf, ps |

9/12 | Review/Overview of (Co)homology | Munkres: Chap 7; Sects 31,33 App. A: pp257-260, 263bot-266, 268 | |

9/14 | |||

9/19 | (Co)homology of Manifolds | Appendix A: pp270-279 | |

9/21 | Applications of Stiefel-Whitney Classes | Section 4: pp37-50 | |

9/26 | Stiefel-Whitney Classes and Cobordism Grassmannians |
Section 4: pp50-53 Section 5: pp55-62; G&H: pp193-194,207 |
pdf, ps |

9/28 | Grassmannians and Vector Bundles | Section 5: pp62-70 | |

10/3 | (Co)homology of CW-Compelexes | Sect. 6: pp73-74; App. A: pp260bot-263 | |

10/5 | (Co)homology of Grassmannians | Sections 6,7; G&H: pp194-197 | |

10/10 | Schubert Calculus (by Paul and Ritwik) | G&H: pp197-206; Vakil's slides | |

10/12 | Thom Isomorphism Theorem |
Spanier pp255-263; Munkres: pp279-280 Sections 8,10; App. A: pp265bot-268top |
pdf, ps |

10/17 | |||

10/19 | Euler Class | Section 9 | pdf, ps |

10/24 | Topology of Smooth Manifolds | Section 11 | |

10/26 | |||

10/31 | Chern Classes | Section 12: pp143bot-146 Sections 13,14 |
pdf, ps |

11/2 | |||

11/7 | Enumerative Geometry of Plane Curves (by Caner and Chris) | expository notes | |

11/9 | Atiyah-Bott Localization Theorem (by Robert F. and Thomas) |
Atiyah-Bott's paper: Sections 2,3 Mirror Symmetry: Sections 4.1-4.3 | |

11/14 | Pontrjagin Classes | Section 15 | pdf, ps |

11/16 | Chern and Pontrjagin Numbers | Section 16: pp183-193 | |

11/21 | Applications to Oriented Cobordism Ring | Section 16: pp190-195; Section 17 | |

11/28 | Some Homotopy Theory | Spainer: Sections 7.2-7.5 Section 18 |
pdf, ps |

11/30 | Thom-Pontrjagin Construction | ||

12/5 | Multiplicative Sequences | Section 19: pp219-226 Warner: Chapter 6 | |

12/7 | Hirzebruch's Signature Theorem | ||

12/12 | Exotic Smooth Structures on S^{7} |
Milnor's paper | |

12/14 | Atiyah-Singer Index Theorem | Spin Geometry: Chapter III |

This page is maintained by Aleksey Zinger.