# Problem Sheet C

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! 🤓

Comments and questions?