# Problem Sheet B

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*

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! 🤓

Comments and questions?