Genus 2 surface with solid disc inside I'm trying to calculate the homology groups of the genus 2 surface with a solid disc inside. I have included a picture 

I'm aware of how to compute the homology groups of a genus $g$ surface, with this disc in there however I'm confused. The method I have tried is to consider the fundamental polygon, the octagon with edges $a,b,a^{-1}, b^{-1}, c,d,c^{-1}, d^{-1}$ and somehow incorporate the disc in there somehow. 
 A: One way is to perform some manipulations which preserve the homotopy type.  The disk can be contracted to a point (since it can be thought of as a CW subcomplex, for instance), and then stretched into an interval, and the resulting space is the wedge sum of a torus and a circle.  The homology groups are (in increasing dimension) are $\mathbb{Z}$, $\mathbb{Z}^3$, $\mathbb{Z}$.
Another way is to compute CW homology directly.  Let $a$ be the perimeter of the disk, and let $b,c,d$ be the other loops in the standard CW composition, such that the fundamental polygon has boundary $aba^{-1}b^{-1}cdc^{-1}d^{-1}$.  Let $B$ the the additional disk, which has boundary $a$.  The first homology group involves an extra relation from $B$, since $\partial B=a$.  So, instead of being generated by $a,b,c,d$, the $B$ makes $a$ nullhomologous, and so $b,c,d$ are the generators of $H_1$.  Since $\partial B\neq 0$, $H_2$ remains the same.
Another way is the Mayer-Vietoris sequence.  Let $A$ be the original surface, and let $B$ be the disk.  For $H_1$, we have that $H_1(A\cap B) \to H_1(A)\oplus H_1(B)$ is injective (in fact, its image in $H_1(A)$ is one of the generators), so $H_1(A\cup B)$ is $H_1(A)/H_1(A\cap B)$, which is $\mathbb{Z}^3$.  For $H_2$, we have $H_2(A\cap B)=0$ and $H_2(B)=0$, so since the following map in the sequence is injective, $H_2(A)\cong H_2(A\cup B)$.
Another way is to recognize that it is homotopy equivalent to collapsing the disk, and then using the fact that the genus-$2$ torus and the boundary of the disk form a good pair, the reduced homology of the quotient is the relative homology of the pair $(T,S)$, with $T$ the genus-$2$ torus and $S$ the boundary of $B$.  The long exact sequence is
\begin{equation}
\widetilde{H}_2(S)\to \widetilde{H}_2(T) \to H_2(T,S)\to
\widetilde{H}_1(S)\to \widetilde{H}_1(T) \to H_1(T,S)\to 0
\end{equation}
which is
\begin{equation}
0\to \mathbb{Z} \to H_2(T,S) \to \mathbb{Z} \to \mathbb{Z}^4 \to H_1(T,S) \to 0
\end{equation}
There isn't much we can do without realizing $\widetilde{H}_1(S)\to\widetilde{H}_1(T)$ is an injection whose image is generated by a generator of $\widetilde{H}_1(T)$, but then we have $H_2(T,S)=\mathbb{Z}$ and $H_1(T,S)=\mathbb{Z}^3$.
