Introduction to algebraic topology
代數拓墣導論
授課教師: 李宗儒 Tsung-Ju Lee
學分數: 3
上課時間: 星期三 1:10pm - 2:00pm, 星期四 9:10am - 11:00am
上課地點:
課程網站:
參考書目：
- Allen Hatcher, Algebraic Topology.
- James W. Vick, Homology Theory: An Introduction to Algebraic Topology.
- Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology.
先修課程或先備能力：基礎代數（能使用群、環、體、向量空間等基礎代數語言）、 基礎拓墣學（高等微積分中講授的點集拓墣知識）。 Students are supposed to be familiar with basics of algebra (languages of groups, rings, fields, vector spaces) and point-set topology (covered in Advanced Calculus).
重要公告與注意事項：
預計課程內容與進度：
| 上課日期 | |
|---|---|
| 2/21, 2/22 | Path and homotopy; fundamental groups. |
| 2/28 (停課), 3/1 | Examples; covering spaces. |
| 3/6, 3/7 | Lifting properties; universal covers. |
| 3/13, 3/14 | Deck transformations; van Kampen theorem (optional). |
| 3/20, 3/21 | Singular homology: ideas and the construction. |
| 3/27, 3/28 | Relative homology; exact sequences; subdivisions. |
| 4/3, 4/4 (停課) | Excision; Mayer--Vietoris sequences. |
| 4/10, 4/11 | Homology and fundamental group; applications. |
| 4/17, 4/18 | CW complexes; cellular homology. |
| 4/24, 4/25 | Midterm; singular cohomology: ideas and the construction. |
| 5/1, 5/2 | Universal coefficient theorem for singular cohomology. |
| 5/8, 5/9 | Examples; cup product and cohomology ring. |
| 5/15, 5/16 | Kunneth formula; Poincare duality. |
| 5/22, 5/23 | Poincare duality; applications. |
| 5/29, 5/30 | de Rham cohomology; Cech cohomology. |
| 6/5, 6/6 | A formal viewpoint; categories and functors. |
| 6/12, 6/13 | Sheaf cohomology. |
| 6/19, 6/20 | Final exam/oral presentation. |