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! 😀