Elizabeth Denne Assistant Professor Office: 312 Burton Hall Telephone: (413) 585 3757 Email: edenne[at]smith [dot] edu

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Essential Information:

Meetings: Mondays 1 - 2pm, Fridays 11am - 12.10pm Course webpage: http://www.math.smith.edu/~edenne/teach/alg-top-studies.html Office hours: 1-3pm Tuesdays, 1-3pm Fridays and by appointment. Office: 312 Burton Hall Phone: 413 585 3757 E-mail: edenne[at]smith [dot] edu Required Textbook: Algebraic Topology, a first course by William Fulton. Graduate Texts in Mathematics vol 153, Springer Verlag 1997. Course reference: Algebraic Topology by Allen Hatcher. Available as a textbook and also as a free download ( found here.) It is not required for the course. Syllabus: From Algebraic Topology, a first course. Calculus and topology: chapters 1, 2. Winding numbers and their applications: chapters 3, 4. Cohomology and homology: chapters 5, 6, 10. Homotopy and covering spaces: chapters 11, 12, 13, 14. Prerequisites: Introduction to Topology and consent of the instructor. Grades: The grade for the course will depend on three things: (1) coming to meetings, (2) asking questions and presenting material during the meetings, and (3) writing up solutions to the problems assigned each week. Drop date: The drop date for the course is Friday September 18.

Reading Assignments:

Date Reading Problems 9/8-9/11 Algebraic Topology Ch 0 pages 1-6 Ch 0 p. 18 ex 1, 2. 9/11-9/18 Fulton Chapter 1 pages 3-16. Add more details to the proof of Prop 1.2, and fill in the details for the example on p 6, 7. Check the criterion 1.10 for this example. What do you notice? You've looked at w_theta and w from ex 1-7. Which of these are closed, exact? Problems 1-1, 1-7, 1-9. Optional 1-20. 9/18-9/25 Fulton Chapter 2 pages 17-27. Optional pages 28-31. Pondering ex 2-3 will help you understand the definition of theta(t) and thinking through ex 2-5 will give another perspective. We'll think more about covering maps and spaces in Ch 11, so view some of 2a as preparatory. Ex 2-10 gets you to think about all the stuff that will be comign in Ch 3 :-). We'll also find path and homotopy lifting from 2b in Ch 11. Problems 2-6, 2-12, 2-15 (Green's Theorem and Prop 1.12 are useful), 2-19. 9/25-10/2 Fulton Ch 3 pages 35-47. Read through 2-16 and 2-17 page 26,27 before starting the chapter. Enjoy Propositions 3-11 and 3-20. This last one is particularly useful. Ponder the ideas contained in exercises 3-7, 3,17, 3-25 (a) (b), and 3-26. Key idea is that the winding number and/or degree can be used to great effect. Problems 3-3, 3-14, 3-19, 3-21. 10/2-10/9 Fulton Ch 4 pages 48-58. Fill in the details in Section 4b, c, and d especially for Prop 4.3, 4.4, 4.21, 4.22, 4.32. Think through exercises 4.7, 4.19, 4.36, 4.37. Exercises 4.5, 4.6, 4.7, 4.15, 4.24, 4.28. 10/9-10/16 Fulton Ch 5 sections a and b Review abstract vector space theory and fill in the details of the description of the zeroth and first Cohomology groups and the boundary map. As for reading. 10/16-10/23 Fulton Ch 5 pages 63-77. Review abstract vector space theory and fill in the details of the description of the zeroth and first Cohomology groups, especially Propn 5.1 and 5.3. Review linear transformations and the boundary map. Make sure you understand the proof of the Jordan curve theorem p.69, especially why im(d) is 1 dimensional. Ex 5.4, Proof that the boundary map preserves scalar multiplication (Lemma 5.6), ex 5.8, 5.16. 10/23-11/6 Fulton Ch 6 pages 78-85. Review abstract group theory and fill in any missing details of the description of the zeroth and first Homology groups. Problem 6.1, Ex 6.5, 6.12, 6.14. Plus others from the Homology work sheet. 11/6-11/13 Fulton Ch 6 pages 78-85. Fulton Ch 10 sections a and b pages 137 - 143. Problems 6.22, 6.24, 6.25, 6.26. 11/13-11/20 Fulton Ch 10 pages 137 - 150. Think about Exercise 10.12 and 10.17. Problems 10.4, 10.5, 10.6, 10.7, 10.13.

Homework Policy:

Homework: Readings and problems to be completed are handed out each week. They are at the heart of the course. Submitting HW: The problems should be turned in to me (or left in my office door mailbox) by 5pm each Friday. (You may also submit an electronic version of the assignment. This should be in .pdf format.) Working together: Working together is the best way to proceed. In fact, just hand in one copy of the problems each week. Keep your grader happy: To make the job of grading easier, please follow the following guidelines: Write your names on your homework. Neat, legible handwriting. The problems should be in the order assigned and with some room available for comments. Staple all pages together.