Spring 2005

MAT4520: Manifolds

MAT4520: Manifolds

Main book:

D. Barden, C. Thomas, An introduction to differential manifolds.

Additional literature:

M. Spivak, A comprehensive introduction to differential geometry. Vol. I.

W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry.

I. Madsen, J. Tornehave, From calculus to cohomology.

Lectures 1-4.

Sections 1.1-1.3, 9.1-9.4.

Lecture 5 (7.02.05).

Submanifolds. Partitions of unity. Embedding theorem for compact manifolds. Tangent space. Tangent bundle.

Sections 1.3, 1.4, 1.5, 2.1, 2.2, 2.3.

Lecture 6 (14.02.05).

Tangent bundle. Vector fields. The Lie bracket of vector fields.

Sections 2.3, 2.5, 2.6.

Exercises: 1.3, 1.4, 1.6 and these two.

Lecture 7 (21.02.05).

Integral curves and 1-parameter groups of diffeomorphisms. Linear algebra: tensor and exterior products.

Sections 2.7, 4.1, 9.7, 9.8.

Exercises: 2.2 and these two.

Lecture 8 (28.02.05).

Linear algebra: tensor and exterior products. Vector bundles.

Sections 4.1, 9.7, 9.8, 3.1, 3.2.

Exercises: these four.

Lecture 9 (7.03.05).

Vector bundles. Cotangent bundle. Differential forms.

Sections 3.1, 3.2, 2.3, 2.5, 4.2.

Exercises: these five.

Lecture 10 (14.03.05).

Differential forms. The orientation of manifolds. Integration of forms.

Sections 4.2-4.4.

Exercises: these four.

Lecture 11 (21.03.05).

The exterior derivative. Manifolds with boundary. Stokes' theorem.

Sections 5.1-5.4.

Exercises: these five.

Lecture 12 (4.04.05).

De Rham cohomology. The Poincare lemma. Approximation of continuous maps by smooth ones.

Sections 6.1-6.3, Exercise 6.1(i).

Exercises: these three.

Lecture 13 (11.04.05).

Cochain complexes. Exact sequences. The Mayer-Vietoris sequence.

Sections 6.2, 6.4.

Exercises: these five.

Lecture 14 (18.04.05).

Cohomology with compact support. Poincare duality.

Chapter 13 in Madsen-Tornehave or Chapter 11 (up to the Thom class) in Spivak.

Exercises: these five.

Lecture 15 (25.04.05).

Applications of cohomology: Brouwer's theorem, the degree of a map and the fundamental theorem of algebra.

Sections 5.4, 5.5, 7.1.

Lecture 16 (2.05.05).

Sard's theorem. Transversality.

Section 9.5.

Exercises: these three.

Lecture 17 (23.05.05 ?).

Exercises: these four.