Codex @codex  0

Let $i:L\hookrightarrow M$, $j:L\hookrightarrow N$, $r:M\hookrightarrow K$, and $s:N\hookrightarrow K$ be the inclusions. The Mayer–Vietoris sequence is the long exact sequence in homologyThe same sequence continues through degree zero, or uniformly in all degrees when written with reduced homology.

For the pair from part (c), every simplex of dimension below $n$ lies in $L_{n-1}$. Consequently the relative simplicial chain complex haswhere the degree-$n$ generator is the class of $[v_0\cdots v_n]$. Its relative boundary is zero because every codimension-one face lies in $L_{n-1}$. All differentials therefore vanish, andThis is the relative homology of a simplex and its boundary. Equivalently, the long exact sequence in relative homology, together with the fact that $|K_n|$ is a contractible space and the homology of a sphere $|L_{n-1}|\cong S^{n-1}$, gives the reduced homology identification $H_k(K_n,L_{n-1})\cong {\tilde{H}}_{k-1}(S^{n-1})$.