Kursöversikt

The official course plan is here.

Lecture 1: submanifolds of R^n, abstract manifolds, smooth structures, charts and atlases, smooth maps. Chapter I.0-I.1

Lecture 2: tangent spaces and the tangent bundle, immersions, submersions, and embeddings. Chapter I.2

Lecture 3: Sard's theorem. Chapter III.1

Lecture 4: More on immersions and embeddings; embedding manfolds in Euclidean space. Chapter I.3

Lecture 5: Tubular neighborhoods (Chapter IV.5 or Bredon Ch II.11) Smooth approximations and applications (Bredon Ch II.11). I am switching the textbook for this chapter to Bredon, Geometry and Topolgy because Hirsch doesn't prove approximation in the way I need it later.

Lecture 6: Transversality (Chapter III.2)

Lecture 7: Transversality (contd) and mod-2 intersection theory (I followed for this Guillemin-Pollack: Differential Topology Ch. II.3-4 but this is also in more general form contained in Hirsch Chapter III.2 and V). Manifolds with boundary (home reading) Chapter I.4

Lecture 8: Morse theory: cotangent bundle, critical points, Hessian, nondegenerate critical points, index, Morse functions, Morse's lemma (Chapter VI.1)

Lecture 9: Morse theory: moving up without passing critical values (Chapter VI.2)

Lecture 10: Morse theory: passing critical values and attaching cells, CW-complexes (Chapter VI.3-4)

Lecture 11: Cobordism theory: oriented manifolds (Chapter IV.4 for the more general concept of an orientation of a vector bundle, or tom Dieck, Algebraic Topology, Ch. 15.5), collars (Chapter IV.6 or tom Dieck, 15.7), the bordism relation, definition of bordism homology (tom Dieck, Ch 21.1).