Interpretation of the connecting homomorphism:

We use equation (29.18) to describe explicitly the connecting homomorphism in the Mayer Vietoris sequence. Take a representative cycle $\zeta$ in $H_n(U\cup V)$. Theorem (34.6) implies that an arbitrary element of $H_n(U\cup V)$ can be represented as a sum of chains

$$\zeta = \zeta_1 + \zeta_2$$

where $\zeta_1\in S_n(U)$ and $\zeta_2 \in S_n(V)$. Note that we are resorting to an abuse notation in writing $\zeta_1$ instead of $i_{\sharp}(\zeta_1)$. We conclude that $\partial \zeta_1 = -\partial \zeta_2$. Thus $\partial \zeta_1$ and $\partial \zeta_2$ are both cycles in $U\cap V$. According to (29.18), the homomorphism $\delta_n$ is given by

$$\delta_n(\overline{\zeta}) = \overline{\partial \zeta_1}$$