Theorem 34.7 (Mayer Vietoris sequence):

(i) Let $\{U, V\}$ be an open covering of $X$,

$$\kappa^{\prime}:H_k(U\cap V)\longrightarrow H_k(U), \quad \kappa^{\prime\prime}:H_k(U\cap V)\longrightarrow H_k(V)$$

be the maps induced by inclusions. Further, let $q_n : H_n(U)\oplus H_n(V)\longrightarrow H_n(U\cup V)$ be the map:

$$(a, b)\mapsto j_{1*}a+j_{2*}b,$$

where $j_{1*}$ and $j_{2*}$ are induced by the respective inclusions $j_1:U\longrightarrow U\cup V$ and $j_2:V\longrightarrow U\cup V$. Then, the following long exact sequence known as the Mayer Vietoris sequence holds:

$$\begin{CD} @> >> H_n(U\cap V) @> (\kappa^{\prime}, -\kappa^{\pri... ... H_n(V) @> q_n >> H_n(U\cup V) @>{\delta_n} >> H_{n-1}(U\cap V) @> >> \end{CD}$$

(ii) A cycle $\zeta \in Z_n(U\cup V)$ may be represented (modulo boundaries) as $\zeta = \zeta_1 + \zeta_2$ for some $\zeta_1\in S_n(U)$ and $\zeta_2 \in S_n(V)$ and the connecting homomorphism $\delta_n$ is given by

$$\delta_n\;:\; \zeta \mapsto \partial \zeta_1 = -\partial \zeta_2.$$