State the Mayer-Vietoris theorem, and use it to calculate, for each integer $q>1$, the homology group of the space $X_{q}$ obtained from the unit disc $B^{2} \subseteq \mathbb{C}$ by identifying pairs of points $\left(z_{1}, z_{2}\right)$ on its boundary whenever $z_{1}^{q}=z_{2}^{q}$. [You should construct an explicit triangulation of $X_{q}$.]

Show also how the theorem may be used to calculate the homology groups of the suspension $S K$ of a connected simplicial complex $K$ in terms of the homology groups of $K$, and of the wedge union $X \vee Y$ of two connected polyhedra. Hence show that, for any finite sequence $\left(G_{1}, G_{2}, \ldots, G_{n}\right)$ of finitely-generated abelian groups, there exists a polyhedron $X$ such that $H_{0}(X) \cong \mathbb{Z}, H_{i}(X) \cong G_{i}$ for $1 \leqslant i \leqslant n$ and $H_{i}(X)=0$ for $i>n$. [You may assume the structure theorem which asserts that any finitely-generated abelian group is isomorphic to a finite direct sum of (finite or infinite) cyclic groups.]