= Solution
For $K=M\cup N$, the simplicial Mayer--Vietoris <sequence> is
$$
\cdots\to H_i(M\cap N)\xrightarrow{(j_*,-k_*)}
H_i(M)\oplus H_i(N)\to H_i(K)
\xrightarrow{\partial}H_{i-1}(M\cap N)\to\cdots.
$$
Construct a simplicial mapping cone of a degree-$k$ map of a circle. Explicitly, let the target circle have vertices $v_0,v_1,v_2$, and let the source circle have vertices $w_0,\ldots,w_{3k-1}$. Map $w_j$ to $v_{j\bmod3}$, triangulate each quadrilateral of its mapping cylinder, and cone the source circle to one new vertex. The resulting finite simplicial complex is the mapping cone $K_k$.
Its reduced cellular, or equivalently simplicial, chain complex collapses to
$$
0\to\mathbb Z\xrightarrow{\times k}\mathbb Z\to0
$$
in degrees two and one. Consequently
$$
H_i(K_k)\cong
\begin{cases}
\mathbb Z,&i=0,\\
\mathbb Z/k,&i=1,\\
0,&\text{otherwise}.
\end{cases}
$$
The same computation follows from Mayer--Vietoris applied to the cone and the mapping-cylinder neighborhood.
Back to article page
