MAT 566 Differential topology

Spring 2009

UNDER CONSTRUCTION

TEXTBOOK:

We will use several sources. The main one is Characteristic classes (Milnor and Stasheff). Others include: Topology from a differentiable viewpoint (Milnor), Differential forms in algebraic topology (Bott and Tu)...

Here is a list of sources that I will update periodically: Topology from a differentiable viewpoint (Milnor), Foundations of differentiable manifolds and Lie groups (Warner), Differential forms in algebraic topology (Bott and Tu), Inroduction to differentiable manifolds and Riemannian geometry (Boothby), Lecture notes on differentiable structures (Milnor collected works volume on differential topology), ...

MEETING TIMES.

Tu+Th 9:50am-11:10am, MAT TOWER 5-127, Instructor: Mark De Cataldo.

PREREQUISITES and COREQUISITES:

The prerequisites for this course are:

  • MAT 531;
  • or permission of the instructor.

    • SYLLABUS.

    We shall cover a (hopefully not too small) subset of the following set of topics.

  • Manifolds, orientation, manifolds with boundary, tangent vectors, derivatives.
  • Brown-Sard theorems, Thom transversality theorem.
  • Brouwer fixed point theorem, Brouwer's theorems on degrees.
  • Vector fields, one-parameter groups of diffeomorphisms, Ehresmann Lemma for proper submersions, Bertini theorem, hairy ball theorem, Poincare'-Hopf.
  • Morse theory, characterization of spheres, Lefschetz hyperplane theorem.
  • Pontryagin framed cobordism theorems, product neighborhood theorem, Hopf theorem on maps to spheres.
  • Vector bundles, tangent and normal bundles, parallelizability, Euclidean vector bundles, basic operations on vector bundles.
  • Review of cohomology.
  • Cellular decomposition of Grassmannians, cohomology ring of Grassmannians, characteristic classes.
  • Stiefel-Whitney classes, Stiefel parallelizability theorem, embedding obstructions, Stiefel-Whitney numbers, Pontryagin-Thom theorems on manifolds being boundaries and on cobordism classes.
  • Euler class, tubular neighborhood theorem, Thom isomorphism theorem, Thom class, diagonal class, relation to Euler characteristic, Lefschetz fixed point theorem, Poincare'-Hopf revisited.
  • Thom spaces, the oriented cobordism ring.
  • Differential forms, Poincare' Lemma, de Rham isomorphism.

    First day of class: TU Jan 27. Last day of class: TH May 7. Spring break April 6-11 (no class on TU+TH April 7-9). 14 Weeks, 28 lectures.

    SPECIAL NEEDS:

    If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services at (631) 632-6748 or http://studentaffairs.stonybrook.edu/dss/. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: http://www.sunysb.edu/ehs/fire/disabilities.shtml

  • Mark Andrea de Cataldo's homepage.
    • Quod non est in web non est in mundo