Background and goals: This course is the Topology part of the qualifying sequence of Math 868-869: Geometry/Topology. The fundamental question that we seek to answer in this course is: how can we tell whether two manifolds are homeomorphic? Over the course of the 20th century, mathematicians have developed a number of algebraic tools to help answer this question. The tools that we will study are the fundamental group (including covering spaces and van Kampen's theorem), homology theory, and some cohomology theory.

Textbook: Agebraic Topology, by Allen Hatcher. We will cover most of Chapters 1 and 2, plus part of Chapter 3.

Prerequisites: a good grounding in undergraduate algebra, plus some background in topological spaces. For example, MTH 310, MTH 411, and MTH 461 would do the trick. This course is almost entirely independent of MTH 868.

Grading: The final grade is based on homework (60%) and a take-home final (40%). Homework will be due weekly, typically on Wednesdays. No late homework is acceptable; however, I will drop your lowest homework grade.

Discussion sections: We will have one weekly discussion section, led by Inanc Baykur. This will focus on examples related to the lectures; sometimes, we'll also go over homework problems. Participation in this seminar is mandatory.

• Homework 1, due 1/17: Page 38 #1, 3, 4. Extra problem: prove that every star-shaped set in R^n is simply connected. (See problem 4 for the definition of star-shaped.)
• Homework 2, due 1/24: Pages 38-39, #8, 9, 10, 16.
• Homework 3, due 1/31
• Homework 4, due 2/7: Pages 53-54, #7, 8, 14, 17.
• Homework 5, due 2/14: Pages 79-80, #4, 9, 11 or 12; additional problem.
• Homework 6, due 2/21: Pages 80-81, #11 or 12 (whichever one you did not do last week); 18, 20, 21.
• Homework 7, due 2/28: Pages 86-87, #5, 7; Page 96, #1, 2.
• Homework 8, due 3/14: Page 131, #1, 4, 5, 8.
• Homework 9, due 3/21: Pages 131-132, #10, 11, 14. Page 53, #9. (Note: you can use the fundamental group to do the last question, but homology makes it easier!)
• Homework 10, due 3/28: Page 132, #16, 17, 19, 22.
• Homework 11, due 4/4: Page 133, #27, 29. Pages 157-158, #28, 32.
• Homework 12, due 4/11: Page 155, #2, 3; additional problems.
• Homework 13, due 4/18: Pages 155-157, #7, 11, 19, 20.

• Take-home final, due 5/2.