Differential Topology by Guillemin and Pollack, Supplemental: Lee's Introduction to Smooth Manifolds

Graduate standing, or consent of instructor

Topics includes: differentiable manifolds, vector fields, transversality, Sard's theorem, examples of differentiable manifolds; orientation, intersection theory, index of vector fields; differential forms, integration, Stokes' theorem, deRham cohomology; Morse functions, Morse lemma, index of critical points.

• (3 lectures) - Basics of smooth manifolds: definitions, atlas, examples, smooth maps, submanifolds, manifolds with boundary (Guillemin Pollack 1.1, 2.1 + Lee Chapter 1)

• (2-3 lectures) - Tangent spaces and derivatives: definitions, examples, chain rule (Guillemin Pollack 1.2)

• (1-2 lectures) - Vector bundles: definitions, tangent bundle, normal bundle, vector fields (Instructor notes, Lee selected portions of Ch 3 and Ch 10)

• (3-4 lectures) - Immersions, Submersions, Regular values: Inverse function, local model for immersion, Implicit function, local model for submersion, regular values preimage theorem and tangent space, Sard’s theorem (omit proof) (Guillemin Pollack 1.3, 1.4)

• (1-2 lectures) Morse theory: Morse’s lemma, genericity of Morse functions (Guillemin Pollack 1.7)

• (3 lectures) Transversality: Definitions, examples, transverse intersection theorem, stability, genericity (skip proof) (Guillemin Pollack 1.5, 1.6, 2.3)

• (3 lectures) Intersection theorem mod 2: definition, invariance, examples, mod 2 degree (Guillemin Pollack 2.4)

• (3-4 lectures) Orientation and oriented intersection: definitions, boundary orientation, preimage orientation, oriented intersection number, invariance, examples, degree (Guillemin Pollack 3.2, 3.3)

• (if time) Index of zeros of a vector field, Euler characteristic, Poincare-Hopf (Guillemin Pollack 3.5)

• (4 lectures) Differential forms: definitions, wedge product, exterior derivative, pull-back, integration, change of variables, de Rham cohomology (Guillemin Pollack 4.2, 4.3, 4.4, 4.5, 4.6)

• (1-2 lectures) Stokes’ theorem: orientation, statement and proof of theorem, applications (Guillemin Pollack 4.7)

If pacing absolutely requires further cuts to the syllabus, reduce time spent on oriented intersection theory, and skip to differential forms. Most of the material should come up in lectures, supplemented with problem sets, but selected portions of the material may also may be taught solely through guided problem sets. If time allows, lectures can add more about vector fields, fixed point theory, other applications of intersection theory (mod 2 or oriented), or other applications of differential forms such as Gauss-Bonnet.