(a) State the Seifert-van Kampen Theorem.
(b) Give a cell decomposition of the Klein bottle $K$. Use this to give a presentation of $\pi_1(K)$.
(c) Let $B$ be a small open ball in $K$. Give a presentation of $\pi_1(K \backslash B)$. What word does the boundary represent in $\pi_1(K \backslash B)$ ?
(d) Let $S$ be a surface obtained by gluing together two copies of $K \backslash B$ along a homeomorphism of their boundaries. Compute $\pi_1(S)$ using the Seifert-van Kampen Theorem.
(e) Prove that $\pi_1(S)$ is not abelian by constructing a surjective homomorphism $\phi: \pi_1(S) \rightarrow S_3$, where $S_3$ is the symmetric group on three letters. Using this, show that there is no covering map from $S$ to the torus.