Problem Sheet D
- 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! 🤓
Comments and questions?