This sheet is based on Lectures 7 and 8.

• Problem 1 asks to prove some fun facts about (free) abelian groups. Yay, group theory! 🙃
• Problem 2 asks you to show that all simplices are homeomorphic to balls. Hint: First prove it for the standard $n$-simplex $\Delta^n$. Then show any two $n$-simplices are homeomorphic via an affine map.
• Problem 3 has a fancy name: this is the dimension axiom. It won't be until the end of the course when we cover the Eilenberg-Steenrod axioms that you understand the name. For now, just think of it as the "axiom that a point has boring homology".
• Problem 4 relates the homology of a space to that of its path components. Note this is different to how the fundamental group behaves (cf. Proposition 4.10).

This is another nice and easy sheet. 🎉

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