# Problem Sheet N

This sheet is based on Lectures 28, 29, 30 and 31.

- Problems 1-2 fill in gaps from Lectures 28 and 29.
- Problem 3 is really long. However despite its length, this problem is not hard, so don't be put off trying to solve it. It shows that divisible groups are “dual” to free groups. (However, if you are pressed for time, I recommend skipping this problem and moving onto the next two, since they are more important.)
- Problem 4 is a useful fact about cup products.
- Problem 5 asks you to compute the cohomology ring for spheres and tori. This problem is meant to be an “exam style” question. 😈

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

Comments and questions?