(a) State a version of the Seifert-van Kampen theorem for a cell complex $X$ written as the union of two subcomplexes $Y, Z$.

(b) Let

$$X_{n}=\underbrace{S^{1} \vee \ldots \vee S^{1}}_{n} \vee \mathbb{R} P^{2}$$

for $n \geqslant 1$, and take any $x_{0} \in X_{n}$. Write down a presentation for $\pi_{1}\left(X_{n}, x_{0}\right)$.

(c) By computing a homology group of a suitable four-sheeted covering space of $X_{n}$, prove that $X_{n}$ is not homotopy equivalent to a compact, connected surface whenever $n \geqslant 1$.