Differential Topology 08/09 (DMat)

Lecturer:	Gustavo Granja
Official course webpage	(does not allow me to input these lecture summaries )
Schedule:	Wed 11-12h30, Room P6 and Fri 11-12h30; Room P7.

Syllabus

Differentiable Manifolds: Differentiable structures. The tangent space. Submanifolds. Embeddings and immersions. Manifolds with boundary.

Approximation and transversality: Topologies on spaces of smooth functions. Approximation by smooth functions. Existence and uniqueness of a smooth structure on a C^1 manifold. Sard's theorem. Transversality.

Vector bundles and tubular neighborhoods: Vector bundles. Basic constructions. Orientations. Classification of vector bundles. Existence and uniqueness of tubular neighborhoods.

Topological invariants: The degree of a map. The intersection number of submanifolds of complementary dimension. The Euler number of a vector bundle.

Morse theory: Morse functions. The Morse lemma. Cell complexes. Morse inequalities. Morse-Bott theory.

Cobordism: The Pontryagin-Thom construction. Thom's theorem.

Depending on the interests and background of the students we might cover other topics such as morse homology or the classification of high dimensional manifolds.

Bibliography

The course will follow Hirsch's book. Other recommended books are:

A. Banyaga and D. Hurtubise, Lectures on Morse homology. Kluwer Texts in the Mathematical Sciences, 29. Kluwer, 2004.
R. Bott and L. Tu, Differential forms in algebraic topology, Springer GTM 82, 1982
V. Guillemin and A. Pollack, Differential topology, Prentice Hall, 1974
M. Hirsch, Differential Topology, Springer GTM 33, 1976
J. Milnor, Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, PUP, 1997.
J. Milnor, Characteristic Classes, Annals of Mathematics Studies, No. 76, PUP, 1974
J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, PUP, 1963
J. Milnor, Lectures on the h-cobordism theorem, PUP, 1965.
J. Munkres, Elementary Differential Topology, Annals of Mathematics Studies, No. 54, PUP, 1963
F. Warner, Foundations of differentiable manifolds and Lie groups, Springer GTM 94, 1983

Here are some corrections and comments on Hirsch's book.
And here is a classification of 1-manifolds which we used in class (this is the solution to a couple of exercises in Hirsch).

See also the Geometry of Manifolds course webpage of MIT's open course ware.

Course webpages from previous years (in portuguese):

Grading policy:

There will be bi-weekly problem sets making up 30% of the grade. Late homework will not be accepted. There will be two tests each counting 35% towards the grade (dates to be arranged). You will be able to make up for one of these tests the week after classes end.

Homework and exams

Homework due 3/10/08: 1.1.1 plus four of the following exercises: 1.1.4,7,8,9; 1.2.5,6,7,8,10,12,14; 1.3.2,3,4,6,7,11,16.
Homework due 17/10/08: Five of the following exercises: 1.4.3,4,5,7,8,9,10,11; 2.1.1,2,3,5,7,9,11,14,15,16.
Homework due 31/10/08: Five of the following exercises: 2.2.1,2,3,4; 2.3.1,4; 2.4.1,3,4,6,7,8,9,10,11,13,14,15,16
Homework due 14/11/08: Five of the following exercises: 3.1.1,2,3,4; 3.2.1-11,13-15
Here is a practice test with solutions.
Here is the test of 18/11/08.
Homework due 28/11/08: Three of the following exercises: 4.1.2,4; 4.2.2,3,5,6,7,8,9; 4.3.1,2,8,9,10.
Homework due 12/12/08: Five of the following exercises: 4.4.2,5,6,8,9,11,12; 4.6.3,4,5,6,7,9; 5.1.1-12.
Homework due 12/1/09: Eight of the following exercises: 5.2.1,2,3,6,9,10,11,17,18,21 ; 6.1,3,4,6; 6.2.3,10,11,13; 6.3.1,2,6,7,8,9; 6.4.1; 7.1.1,3,4; 7.2.1,3,4,5,6,8.
Here is a second practice test with solutions.
Here is the test of 15/01/09.
Here is the make-up exam of 30/01/09.