**Math 520: Differentiable Manifolds I
**

### Basic Information

**Instructor: **Eugene Lerman
**e-mail:** lerman at math dot uiuc dot edu
**Homepage:**
`https://faculty.math.illinois.edu/~lerman`

** Course page:**
`https://faculty.math.illinois.edu/~lerman/520/520syl.html`

**Office:** 336 Illini Hall
**Office Hours:** TBA and by appointment.
**Phone:** 244-9510
**Class meets:** MWF 10 am in 343
Altgeld Hall

### Prerequisites

Point set topology and linear algebra will be very
useful.

If you have any questions or concerns, please contact me by e-mail.

## Course outline

- Manifolds: Definitions and examples including projective spaces and
Lie groups; smooth functions and mappings; submanifolds; Inverse Function
Theorem and its applications including transversality; (co)tangent vectors
and bundles; Whitney Embedding Theorem; manifolds with boundary;
orientations.

- Calculus on Manifolds: Vector fields, flows, and Lie
derivative/bracket; differential forms and the exterior algebra of forms;
orientations again; exterior derivative, contraction, and Lie derivative
of forms; integration and Stokes Theorem.

- Other topics: Sard's Theorem,
Distributions and the Frobenius Theorem; intersection
theory and degree; Lefschetz Fixed Point Theorem; Poincare-Hopf Index
Theorem; DeRham cohomology.

## Text

An Introduction to Differential Manifolds (Paperback) by Dennis Barden and Charles B. Thomas, Imperial College Press; Reprint edition (March 2003).

## Grades

The
course grade will be based on weekly homework (35%),
a midterm (25%) and a final (40%).

lecture of 11/16: Mayer-Vietoris

Last modified: Mon Nov 26 15:06:06 CST 2007