Welcome to the homepage for Math 4220/6220!

This course is a sophisticated look at a very basic question: what are the properties of the sets of solutions of a system of simultaneous equations? If the equations are linear, this question is exhaustively studied in Linear Algebra (MATH 3000), and the answer is very simple: if we have K linearly independant (homogenous) equations on an N dimensional vector space, then the set of solutions is a subspace of dimension N-K.

In this class, we consider the situation where the equations are nonlinear. The solution sets are generally curved and much more interesting, and can have interesting topological and geometric properties as well. But surprisingly, almost all of the time, the set of solutions is still locally identical to a subspace of dimension N-K. Much of the course will be devoted to putting a rigorous foundation under these simple ideas.

This course is an essential prerequisite for students interested in graduate study in geometry and topology. The course introduces the most important properties of smooth manifolds and submanifolds: embeddings and immersions, transversality and intersection theory, and integration, forms, and DeRham cohomology.

## Syllabus

## Lecture Notes and Homework Assignments

- Overview of differential topology
- Fundamental definitions
- Vector calculus
- The Inverse Function Theorem and Immersions
- Submersions and the Preimage Theorem (I)

## Homework 3 Due Wednesday 9/12. - Submersions and the Preimage Theorem (II)
- Transversality (I) (II)
- Homotopy and stability
- Morse theory and embeddings
- Manifolds with Boundary
- Transversal maps are generic (I)

## The (long-awaited) Exam 1 . - Transversal maps are generic (II)
- There is a missing lecture here introducing intersection theory mod 2.
- Intersection theory mod 2
- Winding Numbers and the Jordan-Brouwer Separation Theorem
- The Borsuk-Ulam Theorem

## Homework 4. Due Monday, October 15. - Orientation and Orientability
- The preimage orientation

## Homework 5 (due Wednesday, October 24). - The preimage orientation and the boundary orientation
- There is a missing pair of lectures introducing oriented intersection number.
- The intersection number of a pair of submanifolds
- (Incomplete) Introduction to Lefschetz Fixed Point Theory
- Local Lefschetz numbers
- Vector fields on Manifolds
- Introduction to differential forms
- Differential forms (continued)
- Defining pullback of forms
- Integration of forms
- Differentiation of forms
- Stokes Theorem
- Integration and Degree
- The Gauss-Bonnet Theorem

## The Final Exam. Due next Wednesday at 5pm.