This sheet is based on Lectures 3 and 4.

• Problem 1 explains explains why the space of homotopy classes of map $u \colon I \to X$ is not very interesting, and thus why we need to use relative homotopy classes (i.e. path classes) to obtain a useful invariant.
• Problem 2 generalises Proposition 2.6.
• Problem 3 shows that $\pi_1$ behaves nicely with respect to taking products.
• Problem 4 shows that freely nullhomotopic maps induce trivial maps on $\pi_1$.
• Problem 5 gives an alternative interpretation of $\pi_1(X,p)$: namely as the space of morphisms in the pointed homotopy category from $(S^1,1)$ into $(X,p)$.

This problem sheet is again hopefully not too challenging—next week’s will definitely be harder! 😈

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