Course outline: 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.

Although this course is mostly independent of Math 8061, we will periodically see connections. Many of our examples will be manifolds rather than more general topological spaces. We will see how an orientation on a manifold can be seen algebraically as well as smoothly. Similarly, we will see how the degree of a smooth map can be seen both algebraically and smoothly. Toward the end of the semester, all the threads will converge as we prove the de Rham theorem, relating de Rham cohomology to singular (co)homology. We will also look at Poincaré duality from a smooth point of view.

Textbook: Agebraic Topology, by Allen Hatcher. We will cover most of Chapters 1 and 2, plus part of Chapter 3. On a few occasions, we will also draw on John Lee's Introduction to Smooth Manifolds in order to emphasize connections between the smooth theory from Math 8061 and algebraic topology.

Prerequisites: Some point-set topology and a solid grounding in undergraduate abstract algebra. Math 8061 is "soft" prerequisite.

Homework policy: Homework assignments will be posted on the course webpage, and will typically be due on Wednesdays. At the end of the semester, I will drop your lowest homework score. I encourage you to start early and to discuss the problems with other students. By all means come by my office hours if you have trouble with a problem. The only real caveat to group work is that you must write up your own solutions, in your own words.

As part of the homework, I expect you to have a sufficient grasp of the solutions to provide oral explanations. In practice, this will work as follows: come to my office, at least every other week, ready to discuss at least one problem. On some homework sets, I might mark a problem for oral presentation instead of a written solution.

Final Exam: The final exam will take place on Friday, May 1, from 10:30 to 12:30.