| Week | Date | Topics covered | Practice problems | Assignments | Notes and announcements |
|---|---|---|---|---|---|
| 1 | 1/5 | Overview of the course. Review of metric spaces and topological spaces. |
Get the textbook. | ||
| 1/7 | Reminders of general topology: Discrete and indiscrete spaces. Homeomorphism. Interior and closure. Basis and subbasis. Subspaces, quotients, products. Spheres and disks. | ||||
| 1/9 | More reminders: Coproducts. Separation axioms. Compactness. Connectedness and path-connectedness. | Chapter 0 #12 | |||
| 2 | 1/12 | § 1.1: Homotopy. Homotopy equivalence. Relative homotopy. | Chapter 0 #3, 4, 9, 11 | ||
| 1/14 | § 1.1: Paths. Fundamental group. Induced homomorphism. | § 1.1 #1, 4, 10 | |||
| 1/16 | § 1.1: Change of basepoint. Fundamental group of a product. |
§ 1.1 #2, 3, 6, 11, 14, 15 | |||
| 3 | 1/19 | § 1.1: Fundamental group of the circle. | § 1.1 #5, 7, 12 | ||
| 1/21 | § 1.1: Homotopy lifting property. | HW 1 due 1/21 in class. | Homework solutions are available on OWL, under Resources. | ||
| 1/23 | § 1.1: Some applications. | § 1.1 #8, 9, 16 | |||
| 4 | 1/26 | Basic category theory: Categories and functors. | |||
| 1/28 | More category theory: Natural transformations. Products and coproducts. | § 1.2 #1 | |||
| 1/30 | More category theory: Pullbacks and pushouts. § 1.2: Van Kampen theorem. |
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| 5 | 2/2 | § 1.2: Proof and applications of van Kampen. | § 1.2 #2, 3, 5, 8 | ||
| 2/4 | Chapter 0: Cell complexes. | Chapter 0 #14, 15, 16 | HW 2 due 2/4 in class. | ||
| 2/6 | Double lecture 10:30 - 12:20. Appendix: Topology of cell complexes. § 1.2: Fundamental group of cell complexes. Surfaces. |
Appendix #2, 3 Chapter 0 #17 § 1.2 #7, 9, 15, 16 |
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| 6 | 2/9 | Lecture canceled. | |||
| 2/11 | § 1.3. Covering spaces. | § 1.3 #1, 2, 3, 4, 7 | |||
| 2/13 | Lecture canceled. | ||||
| Feb 16-20 | Reading Week. No lectures. | ||||
| 7 | 2/23 | § 1.3. Existence of covering spaces. | § 1.3 #5, 10 | ||
| 2/25 | § 1.3. Classification of covering spaces. | § 1.3 #9, 14, 15 | HW 3 due 2/25 in class (extended to Thursday by 5 PM). | ||
| 2/27 | Double lecture 10:30 - 12:20. § 1.3. Deck transformations and group actions. § 2.1. Delta-complexes |
§ 1.3 #17, 18, 19 § 2.1 #1, 3 |
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| 8 | 3/2 | § 2.1. Simplicial homology. | § 2.1 #4, 5, 7, 8 | Information about oral presentations. | |
| 3/4 | § 2.1. Singular homology. | § 2.1 #11 | |||
| 3/6 | § 2.1. More on singular homology. Homotopy invariance. |
§ 2.1 #12 | |||
| 9 | 3/9 | § 2.1. Proof of homotopy invariance. Reduced homology. Relative homology. Exact sequences. |
§ 2.1 #13, 14, 15 | Background on exact sequences. | |
| 3/11 | § 2.1. Long exact sequence of a pair. | § 2.1 #16, 17, 18 | |||
| 3/13 | § 2.1. Some applications of homology. Cofibrations and good pairs. |
§ 2.1 #20, 22 Chapter 0 #23, 27, 28 |
HW 4 due 3/13 in class (extended to Monday 3/16 in class). | Deadline to propose two topics for your oral presentation. | |
| 10 | 3/16 | § 2.1. The excision theorem. | § 2.1 #27, 30, 31 | Please schedule an appointment to discuss your presentation topic. | |
| 3/18 | § 2.1. Barycentric subdivision and proof of excision. | § 2.1 #23, 24 | |||
| 3/20 | § 2.1. More on barycentric subdivision. Some corollaries of excision: Relative homology of a good pair. Suspension isomorphism. |
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| 11 | 3/23 | Reduced suspension. § 2.1: More corollaries of excision: Relative homology of a cofibration. Local homology. Invariance of domain. | |||
| 3/25 | § 2.1: Equivalence of simplicial and singular homology. § 2.2. Degree |
§ 2.2 #1, 2, 3, 4, 5 | |||
| 3/27 | § 2.2. More on the degree. Cellular homology. |
§ 2.2 #9, 10 | HW 5 due 3/27 in class (extended to Monday 3/30 in class). | Please fill out the Doodle poll regarding oral presentations. | |
| 12 | 3/30 | § 2.2. More on cellular homology. | § 2.2 #12, 13, 14, 17, 19 | Deadline to submit a presentation outline. | |
| 4/1 | § 2.2. Homology of surfaces Mayer-Vietoris sequence. |
§ 2.2 #28, 29, 31, 32, 33 | |||
| 4/3 | Good Friday. No lecture. | The schedule of presentations and grading information have been posted. | |||
| 13 | 4/6 | § 2.2. Applications of Mayer-Vietoris § 2.B. Jordan curve theorem. |
§ 2.B #1, 2, 3 | ||
| 4/8 | Last lecture. § 2.2. Homology with coefficients. Additional notes. |
§ 2.2 #40, 42, 43 | |||
| 4/10 | No lecture. | HW 6 due 4/10 at the oral presentations (extended to Monday 4/13 by 5pm). | |||
| Friday 4/10 | Presentations I, 5 - 9 PM. | ||||
| 14 | Friday 4/17 | Presentations II, 5 - 10 PM. | |||
| 15 | Tuesday 4/21 | Presentations III, 9:30 AM - 1:30 PM. | Office hour canceled today (Tuesday), moved to tomorrow (Wednesday 3:30 - 5:30). | ||
| Friday 4/24 | Additional office hours 1:30 - 3:30. You can pick up your Homework 6 and your oral presentation comment sheet. | ||||
| Monday 4/27 | Final Exam, 9:00 AM - 1:00 PM, room HSB 9. | ||||