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 is a fibre bundle (and hence also a 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 of a sphere bundle and use it to compute the cohomology ring of $\mathbb{R}P^n$ with $\mathbb{Z}_2$ coefficients. This is an “exam style” question. 😈

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