# Math 519 – Differentiable Manifolds II – Spring 2014

This course is the second part of a sequence of two courses dedicated to the study of the notion of differentiable manifold. In the first course we have seen the basic definitions (smooth manifold, submanifold, smooth map, immersion, embedding, foliation, etc.), some examples (spheres, projective spaces, Lie groups, etc.) and some fundamental results (constant rank theorem, Witney’s embedding theorem, Frobenius theorem, Stokes Formula, etc.). In this course we will proceed with this study, introducing more advanced notions (e.g., vector bundles, principal bundles, connections, etc.) and aiming at a deeper understanding of smooth manifolds. If you have not taken Math 518 the past semester, you should make yourself familiar with the topics study in that course (please see the web page of Math 518).

Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Wed 11:00 AM (or by appointment);
Class meets: MWF 9:00-9:50 PM, 441 Altgeld Hall;
Prerequisites: Math 518 or equivalent.

## Syllabus:

• Differential Forms. Review of differential forms. De Rham Cohomology. Properties of de Rham cohomology: homotopy invariance and Mayer-Vietoris sequence. The de Rham Theorem. Computations in cohomology: triangulations and Euler’s formula, PoincarĂ© Duality, degree of a map, index of a vector field.
• Fiber Bundles. Vector bundles, principal bundles, connections, parallel transport, curvature, Chern-Weyl theory, PoincarĂ©-Hopf Theorem.
• Morse Theory. (this topic will be covered only if time permits).

## Textbooks:

The first two topics are covered in my lecture notes, but the following textbooks are highly recommended.

### Recommended Textbooks:

• R. Bott and L.W. Tu, Differential Forms in Algebraic Topology,Springer-Verlag, GTM vol 82, 1982.
• John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, GTM vol 218, 2003. See the contents and first chapter here.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, (3rd edition) Publish or Perish, 1999.

• Expository Paper Students will be required to write (in LaTeX) and present a paper. The papers will be due by the last week of classes. Presentations will be in class, during the last week of classes.
• Problem Sessions There will be problem sessions every other week. Students are not required to turn in homework, but homework problems will be discussed in these sessions.

## Homework Assignments and Sections covered so far:

• Homework #1 Read Lectures 18 and 19 and solve exercises 18.2, 18.3, 19.1, 19.2, 19.3, 19.4.
• Homework #2 Read Lectures 20 and 21 and solve exercises 20.3, 20.7, 20.8, 20.9, 21.1, 21.4, 21.5, 21.6, 21.7, 21.8.
• Homework #3 Read Lectures 13 and 22 and solve exercises 13.1, 13.4, 13.6, 13.7, 13.8, 22.1, 22.3, 22.6, 22.8.
• Homework #4 Read Lectures 14 and 23 and solve exercises 14.1, 14.3, 14.6, 14.7, 14.8, 14.10, 14.11, 23.2, 23.3, 23.7, 23.8.
• Homework #5 Read Lectures 24-26 and solve exercises 24.3, 24.4, 25.1, 25.3, 25.6, 26.1, 26.2, 26.3, 26.6, 26.8.

Sections of the Lecture Notes covered: Lectures 13-14, 18-30.