Middle East Technical University Department of Mathematics Fall 2004
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Instructor Prof. Sergey Finashin
Syllabus
1. Charts, atlas, differential structure. Smooth maps. Diffeomorphisms.
Submanifolds. Examples.
2. Manifolds with a boundary. The collars. Smoothing of the
corners in the products and after gluing.
3. Tangent and cotangent vectors (3 definitions) and vector bundles. Local triviality,
fibrations and vector bundles.
Smooth vector bundles, operations. Vector fields and
differential forms as sections. Frame fields.
4. Submanifolds. Immersions, submersions, embeddings. Embedding and immersions
of manifolds into an Euclidian space
Whitney embedding theorem for closed manifolds.
5. Orientation of a vector bundle and of a manifold: comparing different definitions. Examples. Boundary, complex manifolds, products, regular value theorem (respecting the orientation).
6. The flow the local flow of a vector field (as 1-parameter group of diffeomorphisms).
Lie derivative for vector fields and k-forms. Properties of the Lie bracket. Lie groups and Lie algebras.
Midterm 1
7. Geodesic flow and the tubular neighborhood theorem.
8. Connected sums and the Morse surgery. An isotopy between embedded balls.
9. Partition of unity and its applications: smooth approximation, etc.
10. Exhausting compacts. Existence of proper functions and Whitney embedding theorem in the non-compact case.
11. Sard’s theorem and applications. Approximation by immersions and embeddings.
12. Degree of a map, its properties. Homogeneity of a manifold. Some examples and applications (the fundamental theorem of algebra, the Brawer fixed point theorem). The degree of a null-cobordant map.
13. Hopf’s theorem about the maps to a sphere. The Borsuk-Ulam theorem. The linking number as a degree.
14. The index of vector fields. Hopf’s theorem (the sum of indices inside a ball). The Poincare-Hopf theorem.
15. The Milnor number of an isolated singularity of a function as the index of its gradient. Stabilization of a singularity.
Examples.
16. Transversality of submanifolds and maps. The intersection of transverse submanifolds and the transverse inverse image
theorem. Orientation.
17. Transversality theorem as a generalization of Sard’s theorem.
18. Intersection forms and linking forms. Examples. Seifert forms for knots.
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Optional chapters (not included
into the exam)
19. The quadratic forms of 4-manifolds. Even and odd forms. Rokhlin’s, Freedman’s and Donaldson’s theorems.
20. The Hopf invariant. The Pontrjagin and the Thom constructions. Stable homotopy groups of spheres and cobordism
groups.