This sheet is based on Lectures 5 and 6.

• Problem 1 asks you to consider another categorical concept: the coproduct. You are asked to prove that if it exists, then it is necessarily unique. Then you get to prove that in $\mathsf{Groups}$  it does indeed exist.
• Problem 2 allows you compute $\pi_1$ for a "bouquet" of circles (like a bouquet of flowers, but less romantic to non-mathematicians.)
• Problem 3 asks you to compute $\pi_1$ for three spaces that one can obtain by identifying sides of the unit square. This problem is really important—if you only do one problem, make it this one! It serves as a good example as how to use the Seifert-van Kampen Theorem to compute things.
• Problem 4 asks you to prove the Fundamental Theorem of Algebra using topological methods. (Chances are you've already proved this result in at least three other courses you've taken.)
• Problem 5 gives a criteria for $\pi_1$ to be abelian. A key example of a type of space satisfying these hypotheses are topological groups—these are groups $G$ endowed with a topology such that the multiplication map $G \times G \to G$ and the inverse map $G \to G$ are both continuous. The circle $S^1$ is a simple example of a topological group. Thus the fundamental group of any topological group is abelian.

This problem sheet is more difficult than the last two. 😈

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