• Problem 1 asks you to check that in the definition of a smooth map we can replace “every chart” with “any chart”.
  • Problems 2-5 ask to you to prove that various “common” spaces are smooth manifolds.
  • Problems 6-7 give two examples of spaces that fail to be manifolds.
  • Problem 8 shows that a manifold can admit more than one smooth structure. (I will say a bit more about diffeomorphism classes next lecture).
  • Problem 9 is hard. (I do not realistically expect anyone to do this!)

Comments and questions?