Consider the surface $M_g$ of genus $g$, embedded in $\Bbb{R}^3$ in the standard way. It bounds some compact region $R$. Two copies of $R$ are glued together by the identity map between their boundary surfaces, which forms a closed 3-manifold $X$. I am asked to compute the homology groups of $X$.
Now the computation of this for $n = 3$ of $H_n(X)$ is straightforward from Mayer - Vietoris. However now for $n=2$, I run into trouble: I am looking at the following part of the LES from Mayer - Vietoris. Put $A$ = one copy of $R$, $B$ = the other copy, their intersection $A \cap B = M_g$. $i,j$ are the inclusion maps of $A \cap B$ into $A$ and $B$ respectively. The lower end of Mayer - Vietoris looks like
$$0 \rightarrow \tilde{H}_2(X) \rightarrow \tilde{H}_1(A \cap B) \stackrel{(i_\ast,j_\ast)}{\longrightarrow} \tilde{H}_1(A) \oplus \tilde{H}_1(B) \rightarrow \tilde{H_1}(C) \rightarrow 0 \rightarrow 0 \rightarrow 0 \rightarrow 0.$$
Now I know that $\tilde{H_1}(A \cap B) = \Bbb{Z}^{2g}$ and the same for $ \tilde{H}_1(A) \oplus \tilde{H_1}(B)$, but this does not imply that $(i_\ast,j_\ast)$ is an isomorphism. The first fact on the homology of $A \cap B$ comes from the CW - structure of $M_g$ having $2g$ one cells, the second from the fact that $A$ and $B$ respectively can be thought of the wedge sum of $g$ tori, which is homotopy equivalent to a wedge of $g$ circles.
Now the complete the problem I need to know the kernel of the map $\Phi$. To do this, I need to know what are
1. Generators for the homology of $M_g$.
2. Generators for the homology of $A$ and $B$.
How do I go about finding these? I would say my main problem in general is making connections between algebraic things and generators for homologies that comes from topology.