Course outline: This course will be an introduction to the geometry and topology of smooth manifolds. We will begin the fall semester with the definitions: what does it mean for a space to (smoothly) look just like Rⁿ? We will go on to study vector fields and differential forms (a way to take derivatives and integrals on a manifold).

The continuation of this course, Math 8062, will study the interplay between the geometry of a manifold and certain ideas from algebraic topology. We will review the idea of the fundamental group and introduce homology - and then relate these algebraic notions to the underlying smooth geometry. We will finish with the de Rham and Poincare duality theorems, which neatly tie together homology, cohomology, and the algebra of differential forms.

Textbook: Introduction to Smooth Manifolds (Second Edition) by John Lee. In addition to the bookstore, the book is available as a PDF download through the Temple library.

We will cover parts of Chapters 1-16, skipping Chapters 7 and 13. Those chapters will likely seem like too much material for one semester, and they are. I believe Lee's book is an excellent reference, but is too encyclopedic to be covered in a linear fashion. We will bounce around to some degree.

To get a better intuitive sense of the topics that we'll cover, as well as the order in which we'll cover them, it helps to look at Chapters 1-8 of A Comprehensive Introduction to Differential Geometry (volume 1) by Michael Spivak. The latter book is not required, but makes a pretty good companion source.

Prerequisites: Concepts of analysis (Math 5041-42). The spring semester course will also rely on abstract algebra (Math 8011).

Homework policy: Homework assignments will be posted on the course webpage, and will typically be due on Wednesdays. No late homework will be accepted, but 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.

Final Exam: The take-home final exam will be distributed on December 12 and due December 15.