# Problem Sheet K

This sheet is based on Lectures 21, 22, and 23.

- Problems 1,2 and 3 are about
*lens spaces*$L(p,q)$. These are interesting three-dimensional manifolds that are can be visualised by gluing together two solid tori via a homeomorphism of their boundaries. They are the simplest class of closed manifolds whose homotopy type does*not*determine them up to homeomorphism. More relevantly for you, they present an excellent opportunity to see the Cellular Boundary Formula (Theorem 20.11) in action. - Problem 4 is easy.
- Problem 5 asks you to prove the famous
*Yoneda Lemma*. This is not as scary as it looks! - Problems 6 shows that the additivity axiom for a homology theory is only relevant for infinite disjoint unions.
- Problem 7 asks you to show that the Mayer-Vietoris long exact sequence holds for an arbitrary homology theory.

As usual, feel free to ask questions if you are stuck! ðŸ˜€

Comments and questions?