Week 1

  • Show that the definition of smooth manifolds in Milnor and Stasheff agrees with the usual definition of smooth manifolds using smooth cordinate charts.
  • Let $(U,V,h)$ be local parametrization of a smooth manifold $M$ of dimension $n$ at $x$ and $h(u)=x$. Show that if $v$ is a linear combination $\partial h/\partial u_i$, $i=1,...,n$ then $v$ is a tangent to $M$ at $x$.
  • Problem 1-A from Milnor and Stasheff
  • Problem 1-B from Milnor and Stasheff
  • Problem 1-C from Milnor and Stasheff

    Week 2

  • Show that local trivialization holds the map $\pi:DM\to M$ for $M$ a smooth manifold. This completes the definition of tangent bundle of a smooth manifold.
  • Problem 2-B from Milnor and Stasheff
  • Problem 2-C from Milnor and Stasheff
  • Problem 2-D from Milnor and Stasheff
  • Problem 2-E from Milnor and Stasheff

    Week 3

  • Problem 3-A from Milnor and Stasheff
  • Problem 3-C from Milnor and Stasheff
  • Problem 3-D from Milnor and Stasheff
  • Problem 3-E from Milnor and Stasheff
  • Problem 3-F from Milnor and Stasheff

    Week 4 and 5

  • Problem 4-A from Milnor and Stasheff
  • Problem 4-B from Milnor and Stasheff
  • Problem 4-C from Milnor and Stasheff
  • Problem 4-D from Milnor and Stasheff
  • Problem 4-E from Milnor and Stasheff
  • Problem 5-A from Milnor and Stasheff
  • Problem 5-B from Milnor and Stasheff
  • Prove Theorem 5.6 from Milnor and Stasheff

    Week 6 and 7

  • Problem 6-B from Milnor and Stasheff
  • Problem 6-C from Milnor and Stasheff
  • Problem 6-D from Milnor and Stasheff
  • Problem 6-E from Milnor and Stasheff
  • Problem 7-A from Milnor and Stasheff
  • Problem 7-B from Milnor and Stasheff
  • Problem 7-C from Milnor and Stasheff

    Week 7