(i) State, but do not prove, the Mayer-Vietoris theorem for the homology groups of polyhedra.

(ii) Calculate the homology groups of the $n$-sphere, for every $n \geqslant 0$.

(iii) Suppose that $a \geqslant 1$ and $b \geqslant 0$. Calculate the homology groups of the subspace $X$ of $\mathbb{R}^{a+b}$ defined by $\sum_{i=1}^{a} x_{i}^{2}-\sum_{j=a+1}^{a+b} x_{j}^{2}=1$.