U)$ we find $$\begin{aligned}
        H_*(M,\partial M \cup {\Sigmait}) \cong H_*(M_0 \setminus {\Sigmait},\partial M_0 \cup U_-) \oplus H_*(M_1 \setminus {\Sigmait},\partial M_1 \cup U_+),
        \end{aligned}$$ where $U_-$ and $U_+$ is the image of ${\Sigmait}\times (-1,0)$ and ${\Sigmait}\times (0,1)$ in $U$, respectively. Since the inclusion $(M_0 \setminus {\Sigmait},\partial M_0 \cup U_-) {\longrightarrow}(M_0,\partial M_0 \cup U_- \cup {\Sigmait})$ induces an isomorphism in homology and since $\Sigmait {\longrightarrow}U_-\cup \Sigmait$ is a homotopy equivalence, we obtain $ H_*(M_0 \setminus {\Sigmait},\partial M_0 \cup U_-)\cong H_*(M_0,\partial M_0)$ and analogously $ H_*(M_1 \setminus {\Sigmait},\partial M_1 \cup U_+) \cong  H_*(M_1 ,\partial M_1 )$. Thus, we are left with $$\begin{aligned}
     H_*(M,\partial M \cup {\Sigmait}) \cong H_*(M_0,\partial M_0) \oplus H_*(M_1 ,\partial M_1 ).\label{eqngluing2}\end{aligned}$$

Using and we obtain from the exact sequence $$\begin{aligned}
 0 {\longrightarrow}H_{n}(M,\partial M){\longrightarrow}H_{n}(M_0, \partial M_0) \oplus H_{n}(M_1, \partial M_1){\longrightarrow}H_{n-1}({\Sigmait}, \partial {\Sigmait}) \ \    , \end{aligned}$$ where the morphism $H_{n}(M_0, \partial M_0) \oplus H_{n}(M_1, \partial M_1){\longrightarrow}H_{n-1}({\Sigmait}, \partial {\Sigmait})$ takes $[\nu_1]\oplus [\nu_2]$ to $[\partial \nu_1 +\partial \nu_2]$. Evaluating the kernel of this morphism yields an isomorphism $H_{n}(M,\partial M)\cong {\mathbb{Z}}([\nu_1]\oplus [\nu_2])$. This shows that $\nu_0+\nu_1$ is a generator of $H_{n}(M,\partial M)$, i.e. an orientation of $M$. This orientation agrees with the orientation of $M$ in a neighbourhood of an arbitrary point away from 