# Math 109c: Differential Topology, Spring 2012

This class meets MWF 10-11am in 257 Sloan.

**Office hours:** Thursdays 3-5pm in 374 Sloan

**Course Description:** This course will follow Guillemin and Pollack's book Differential Topology. We will study properties of smooth maps (including Sard's theorem and Whitney's "easy" embedding theorem), transversality and intersection theory (including the hairy ball theorem and more generally the Poincare-Hopf theorem), and differential forms (including de Rham cohomology if time permits).

**Errata in G&P:** [1]

# Exercises

There will be a problem set due every Friday at 4pm. You must submit your own solutions, but you are welcome to discuss the problems with others.

Due date | Homework | Description |

Apr. 6 | HW1 | Read 1.1 and 1.2 of G&P. Do problems 3, 4, 6, 18 in 1.1 and 2, 4 (use chain rule), 6, 9 in 1.2. |

Apr. 13 | HW2 | Do problems 7 and 11 of 1.2, problems 1, 2, 3, 4, 5, and 6 (for (a), use exercise 9 of 1.2) of 1.3, and problem 1 of 1.4. Also determine if $\{(x,|x|)\}$ is a manifold. |

Apr. 20 | HW3 | Do the following problems from Guillemin and Pollack: 6 and 7 of 1.4, 5, 7, and 10 of 1.5, 1, 2, 3, and 4 of 1.6, and 5 of 1.7. |

Apr. 27 | HW4 | Do the following problems from Guillemin and Pollack: 13, 16, 17, and 18, (these three are easy) of section 1.7, and 6, 10, (what is the purpose of h in the proof of the theorem on p. 51?) and 15 of section 1.8. |

May 4 | Midterm | You may use any 4 hour interval to complete the exam. You may refer to your class notes, homework, and results proven in the book freely, but do not use any other resources. If you find a typo or don't understand a question, please contact me (email and phone number are in the pdf). |

May 11 | HW5 | Do the following problems from Guillemin and Pollack: 8 of 2.1, 3 and 7 of 2.2, 6, 16, and 20 of 2.3, and 12 of 2.4. |

May 18 | HW6 | Do the following problems from Guillemin and Pollack: 1, 5, 6, 13, and 17 of 2.4, 3 of 2.6, 2 and 9 of 3.2. |

May 25 | HW7 | Do the following problems from Guillemin and Pollack: 12 of 3.2, 2, 4, 10, and 13 of 3.3, 4 and 8 of 3.4. |

June 1 | HW8 | Do the following problems from Guillemin and Pollack: 5 of 3.4, 1, 7, and 19 of 3.5, 1 and 5 of 4.2. Also read section 3.6 and all the exercises in it. |

June 8 | HW9 | Do the following problems from Guillemin and Pollack: 3, 6, and 10 of 4.2, and all the problems in sections 4.3 and 4.5. |

June 15 | Final | You may use any 4 hour interval to complete the exam. You may refer to your class notes, homework, and results proven in the book freely, but do not use any other resources. If you find a typo or don't understand a question, please contact me (email and phone number are in the pdf). |