Differential Topology

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  • What is Sard's Theorem?
  • Give an application of Sard's Theorem.
  • Give a smooth map from S3 to S3. Can "most" points have an infinite number of preimages?
  • Define the Lie bracket of two vector fields on a {\mathbb C}^{\infty} manifold. (Casson)
  • What does it mean to compose two vector fields? i.e. what does XY − YX mean? (Casson)
  • Define vector field in terms of the ring {\mathcal F} of {\mathbb C}^{\infty} functions {{\mathbb M}}\to\R. What does it mean to compose two vector fields? (Casson)
    • Is XY necessarily a vector field?
    • Why is [X,Y] a vector field?
  • When does a vector field determine a flow? (Casson)
  • What does it mean for a vector field to have compact support? (Casson)
  • Define flow. (Casson)
  • In what sense do the diffeomorphisms in a flow vary "in a {\mathbb C}^{\infty} fashion"? (Casson)
  • Does a flow determine a vector field? (Casson)
  • Give conditions on {\mathbb M} so that every vector field on {\mathbb M} determines a flow. (Casson)
  • Relate "tangent vector to a curve at a point" to "point derivation". (Casson)
  • Give an example of a vector field on a manifold that does not determine an everywhere-defined flow. (Casson)
  • A knot is a {\mathbb C}^{\infty} embedding S^1\to \R^3. Consider the following two statements about two knots f,g\colon S^1\to \R^3:
    1. There is an isotopy between f and g.
    2. There is a diffeomorphism of \R^3 inducing a diffeomorphism f(S^1)\to g(S^1).
    Relate these conditions. (Casson)
    • Can you use any of this information to say something about classifying diffeomorphisms \R^3\to\R^3? [Hint: the trefoil knot cannot be deformed into its mirror image] (Casson)
  • How many components does {\rm Diff}(\R^3,\R^3) have, and what is meant by this? (Casson)
  • State a Lemma about a diffeomorphism f\colon \R^3\to \R^3 if f(0) = 0; in particular, how may f be rewritten? (Casson)
  • Write down a path from f to Dif | 0, where f(0) = 0 and f\colon \R^3\to\R^3 is a diffeomorphism. (Casson)
  • Define a Morse function. Define index. (Casson)
  • What is h − 1(a,b) if (a,b) does not contain any critical values? What if it contains exactly one critical value? (Casson)
  • If a morse function on a manifold M has exactly two critical points, what can you say about M? (Casson)
  • What is the Frobenius Integrability Theorem? (Serganova)
  • What is an integral submanifold? (Serganova)
  • What is [x,y]? (Serganova)
  • Can you give an example of a distribution which is not integrable? (Serganova)
  • Explain how the characteristic classes of a vector bundle arise. What are all the characteristic classes of a vector bundle? (Wodzicki)
  • What is the Thom isomorphism? (Wodzicki)
  • What is the Thom class? What is it an obstruction to? (Wodzicki)
  • Construct the Thom class explicitly for a trivial bundle. (Frenkel)
  • Prove that the cohomology of a compact Lie group is that of its Lie algebra. (Wodzicki)
  • How does one use Sard's theorem to prove the Whitney embedding theorem? (Harrision)

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