Topic outline

  • Instructor: Greg Arone,

    General description: Differential topology is the study of manifolds, focusing on their global, topological properties, rather than local, analytic ones. The subject breaks into sub-areas: three-dimensional, four-dimensional, and “high”-dimensional manifolds, which usually means dimensions 5 and above. In this course we will mostly focus to the latter. The goal is to learn the major techniques for working with high-dimensional manifolds: the Whitney trick, surgery, cobordism, and some of their classic applications: the h-cobordism theorem, the high-dimensional Poincare conjecture, exotic smooth structures on high-dimensional spheres. Depending on time and interest, I hope to also make excursions into more recent developments, such as embedding calculus and the stable homology of mapping class groups.

    Here is a somewhat more detailed list of topics that I hope to cover:

    -Review of smooth manifolds. Submanifolds and fibre bundles. Tubular neighbourhoods. The isotopy extension theorem. Boundaries and corners.
    -General position and transversality.
    -The theory of handle decompositions.
    -The h-cobordism theorem and applications. High dimensional Poincare conjecture.
    -Immersions and embeddings. The Whitney trick. Whitney’s embedding theorem. Smale-Hirsh theory. A possible excursion into Embedding calculus.
    -Surgery. Homotopy theory of Poincare complexes.
    -The Thom-Pontryagin construction. The cobordism ring. 
    -Exotic smooth structures. Groups of homotopy spheres.

    Prerequisites: I will begin with a quick review of basic notions about manifolds, but I assume that people saw this stuff before. I will expect some familiarity with algebraic topology, at least on the level of homology and cohomology, as well as the definition of homotopy groups. Some familiarity with more advanced notions of homotopy theory is desirable. For example: fibrations and cofibrations, simplicial sets, classifying spaces. But I will try to review such concepts when needed.

    Textbook: For the first few weeks we will mostly follows the book Introduction to smooth manifolds, by John Lee. For the more advanced material, I am planning to follow quite closely the book Differential Topology by C.T.C. Wall. Both books are available electronically via the Stockholm University library.

    Accreditation: Those who wish to get credit for the course can do it by giving one or two presentations, on a topic agreed with the instructor. Another possibility is to take an oral exam at the end of the course.

    First lecture: Thursday, Jan 25, 10-12, Cramer room.
    Proposed class time: Thursday 10-12, Cramer room.

  • Lecture 1, Jan 25

    Review of the definition of a smooth manifold and some basic properties. Smooth functions and smooth maps. Paracompactness. Partitions of unity.

    Recommended reading: Lee, Chapters 1 (except for the manifolds with boundary) and 2.

  • Lecture 2, Feb 1

    Manifolds with boundary. Tangent vectors and the tangent bundle. The differential of a smooth map. The definitions of immersions, submersions and embeddings. Proved that every manifold can be embedded in R^N for some N. (Only proved the case of compact manifolds in class. Interested students can read the proof of the general case in Lee, Proof of Theorem 6.15).

    Recommended reading: Lee, Chapter 1, the section about manifolds with boundary. Then Chapters 3-5, and the proof of Theorem 6.15

  • Lecture 3, Feb 8

    The rank theorem. Critical values. Sard's theorem.

    Recommended reading: Lee Chapter 4, and first half of chapter 6.

  • Lecture 4, Feb 16

    Finish the proof of the Whitney's "easy" embedding theorem: previously we proved that every manifold M can be embedded in some Euclidean space. Today we use an easy case of Sard's theorem to prove that M can be embedded in space of dimension 2 dim(M) + 1 (and immersed in dimension 2 dim(M))

    Transversality. The preimage theorem for transversal maps. The density of transversal maps.

    (Time permitting) normal bundles and tubular neighbourhoods.

    Recommended reading: Lee, chapter 6.