(a) State the Mayer-Vietoris theorem for a union of simplicial complexes

$$K=M \cup N$$

with $L=M \cap N$.

(b) Construct the map $\partial_{*}: H_{k}(K) \rightarrow H_{k-1}(L)$ that appears in the statement of the theorem. [You do not need to prove that the map is well defined, or a homomorphism.]

(c) Let $K$ be a simplicial complex with $|K|$ homeomorphic to the $n$-dimensional sphere $S^{n}$, for $n \geqslant 2$. Let $M \subseteq K$ be a subcomplex with $|M|$ homeomorphic to $S^{n-1} \times[-1,1]$. Suppose that $K=M \cup N$, such that $L=M \cap N$ has polyhedron $|L|$ identified with $S^{n-1} \times\{-1,1\} \subseteq S^{n-1} \times[-1,1]$. Prove that $|N|$ has two path components.