Problem Sheet K

This Problem Sheet is based on Lectures 20 and 21.

• Problem 1 gives examples of manifolds that are (not) orientable.
• Problem 2 explains why our preferred half-space is “better” than the standard upper half-space.
• Problem 3 is an introduction to the best area of mathematics.
• Problem 4 shows that the product of two smooth manifolds with boundary can fail to be a smooth manifold with boundary.
• Problem 5 is a joke. Seriously. Don’t do it.