This sheet is based on Lectures 43, 44, and 45. This is the last Problem Sheet of the entire course! 🎉

• Problem 1 gives another algebraic proof that $\pi_n(X)$ is abelian for $n \ge 2$.
• Problem 2 shows that any map can be "replaced" by a fibration.
• Problems 3 and 4 are about the long exact sequence of homotopy groups for (weak) fibrations.
• Problem 5 gives an example of two spaces whose homology groups are the same, but whose homotopy groups are not. This is an “exam style” question.

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