This sheet is based on Lectures 36, 37, 38, and 39.

  • Problem 1 considers the homology groups $H_n(M, M \setminus K ;R)$ when $M$ is not $R$-orientable along $K$.
  • Problem 2 shows that for a manifold of dimension $n$, one can also obtain information about the torsion subgroup of $H_{n-1}(M, M \setminus K)$.
  • Problem 3 asks to check that the cap product $ \frown$ really does descend to (co)homology.
  • Problem 4 asks you to prove the homotopy axiom for Čech cohomology.
  • Problem 5 asks you to prove Alexander Duality. This is a massive generalisation of the Jordan-Brouwer Separation Theorem we proved in Lecture 17.

🤓 Feel free to ask a question if you are stuck! 🤓

Comments and questions?