The surface $M_g$ of genus $g$ (i.e. a connect sum of $g$ tori), when embedded in $\mathbb{R}^3$, bounds a compact solid region $R \subseteq \mathbb{R}^3$. Let $X$ be two copies of this space $R$ with their boundaries (the actual surface) identified via the identity map, giving a three-manifold. We calculate the homology. Let $A$ be one region with an open neighborhood into the other, and likewise for $B$. The region $R$ bounded by $M_g$ deformation retracts onto its core -- a wedge of $g$ circles. So, $A$ and $B$ deformation retract onto a wedge of $g$ circles.

$$A \cap B$$ deformation retracts onto $$M_g$$ itself. So $$\wt{H}_1(A) = \wt{H}_1(B) = \mathbb{Z}^g$$, $$\wt{H}_1(A \cap B) = \mathbb{Z}^{2g}$$ and $$\wt{H}_2(A \cap B) = \mathbb{Z}$$ (and others zero). Meyer-Vietoris gives:$$

0 \rightarrow \wt{H}_3(X) \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow \wt{H}_2(X) \rightarrow \mathbb{Z}^{2g} \xrightarrow{\Phi} \mathbb{Z}^g \oplus \mathbb{Z}^g \rightarrow \wt{H}_1(X) \rightarrow 0

$$First note $$\wt{H}_3(X) \cong \mathbb{Z}$$. Now the details of the map $$\Phi$$. $$\mathbb{Z}^{2g}$$ is generated by the $$a_i$$ and $$b_i$$ in the familiar representation as a polygon, i.e. $$\la a_i, \dots, a_g, b_i, \dots, b_g \ra$$. $$\mathbb{Z}^g \oplus \mathbb{Z}^g \cong \mathbb{Z}^{2g}$$ has the core circles as generators in both summands, so write it as $$\la c_1, \dots, c_g, d_1, \dots, d_g \ra$$. For each 'sub-torus' of the genus $$g$$ surface, the deformation retraction takes one of $$a_i$$ or $$b_i$$ *to* the core circle (in both regions), and contracts the other onto a single point. Say the $$a_i$$'s are mapped onto the core circle. With these assumptions, we have $$\Phi(a_i) = c_i + d_i$$ and $$\Phi(b_i) = 0$$. Firstly, we have $$\wt{H}_2(X) \cong \ker \Phi = \la b_1, \dots, b_g \ra \cong \mathbb{Z}^g$$. Secondly, we compute $$\wt{H}_1(X) \cong \mathbb{Z}^{2g} / \im \Phi$$. We have:$$

\wt{H}_1(X) = \dfrac{\la c_1, \dots, c_g, d_1, \dots, d_g \ra}{\la c_1 + d_1, \dots, c_g + d_g \ra} = \dfrac{\la c_1 + d_1, \dots, c_g + d_g, d_1, \dots, d_g \ra}{\la c_1 + d_1, \dots, c_g + d_g \ra} \cong \mathbb{Z}^g

$$