# Problem Sheet O

This sheet is based on Lectures 32, 33, 34, and 35.

- Problems 1 asks you to show the “twist” map is a natural chain equivalence.
- Problem 2 asks you to prove the
*Hopf fibration* - Problem 3 gives you an explicit generator of $H_n( \mathbb{R}^n, \mathbb{R}^n \setminus 0)$.
- Problems 4 and 5 are about relative cup and cross products.
- Problem 6 asks you to derive the
*Gysin Sequence*

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

Comments and questions?