## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 215B: Topology

**Approved:**2009-05-01, Dmitry Fuchs, Greg Kuperberg

**Units/Lecture:**

Winter, alternate years; 4 units; lecture/term paper or discussion section

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Algebraic Topology, Allen Hatcher, Cambridge Univ, ($30), Dmitry Fuchs' handouts

Search by ISBN on Amazon: 0521795400

Search by ISBN on Amazon: 0521795400

**Prerequisites:**

Graduate standing or consent of instructor.

**Course Description:**

Fundamental group and covering space theory. Homology and cohomology. Manifolds and duality. CW complexes. Fixed point theorems.

**Suggested Schedule:**

Lectures | Sections | Topics/Comments |
---|---|---|

1 lecture | Ch. 2 | Motivation: Stokes' theorem and homology |

Week 1 | Sec. 2.1 | Singular homology; homology of a point and a wedge |

Week 2 | Sec. 2.1 | Chain complexes and homology, chain maps and homotopy invariance |

Week 3 | Sec. 2.1 | Exact sequences, 5-lemma, relative homology, homology sequence of a pair |

Week 4 | Sec. 2.1 | The excision/collapse theorem for good pairs, proof using refinements |

Week 5 | Sec. 2.2 | Homology of spheres, bouquets, and suspensions. Homology of CW complexes. |

Week 6 | Sec. 2.3 | Eilenberg-Steenrod axioms, uniqueness, singular cubic theory |

Week 7 | Ch. 4.1, 4.2 | The Hurewicz and Whitehead theorems. |

Week 8 | — | The Lefschetz fixed point theorem, geometric applications of Euler and Lefschetz numbers |

Week 9 | Ch. 3.4 | Tensor and torsion products, homology with coefficients, universal coefficent theorem, Kunneth formula |

**Additional Notes:**

The pacing is approximate; there is an extra week which should be added to the existing topics list.