State the Mayer-Vietoris Theorem for a simplicial complex $K$ expressed as the union of two subcomplexes $L$ and $M$. Explain briefly how the connecting homomorphism $\delta_{*}: H_{n}(K) \rightarrow H_{n-1}(L \cap M)$, which appears in the theorem, is defined. [You should include a proof that $\delta_{*}$ is well-defined, but need not verify that it is a homomorphism.]

Now suppose that $|K| \cong S^{3}$, that $|L|$ is a solid torus $S^{1} \times B^{2}$, and that $|L \cap M|$ is the boundary torus of $|L|$. Show that $\delta_{*}: H_{3}(K) \rightarrow H_{2}(L \cap M)$ is an isomorphism, and hence calculate the homology groups of $M$. [You may assume that a generator of $H_{3}(K)$ may be represented by a 3 -cycle which is the sum of all the 3 -simplices of $K$, with 'matching' orientations.]