This sheet is based on Lectures 9 and 10.

• Problem 1 asks to you show that the abelianisation of a group can be defined by a universal property, and then asks you to verify that it is indeed given by quotienting out the commutator subgroup.
• Problem 2 asks you to prove that the Hurewicz map is natural. The precise meaning of the word "natural" will be explained later on in the course when we study natural transformations (which are morphisms of functors).
• Problems 3 and 4 are both examples of how to "cancel" things out in homology. Warning: These results do not follow directly from what we did in class, since we not assuming each path is closed.
• Problem 5 shows that the definition of a short exact sequence of chain complexes is not nonsensical.
• Problem 6 is a generalisation of Problem 4 on Problem Sheet D.

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