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Let $M$ be a (connected) compact orientable 3-manifold whose boundary $\partial M$ is homeomorphic to $T^2$ (the torus).
Now consider the solid torus $S=S^1\times D^2$ and choose a homeomorphism $f:\partial S\to\partial M$, then form the adjunction space $X=M\cup_f S$.
Suppose we know the homology of $M$. How can we compute the homology of $X$?

One immediately observes that $X$ is orientable, so that $H_3(X)=\mathbb{Z}$. However the Mayer-Vietoris sequence given by the decomposition $X=M\cup_f S$ is not sufficient to compute $H_1(X)$ and $H_2(X)$. Is there some duality theorem which helps us in this task?

What can be said in the nonorientable case?

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Note that you have a collar of $\partial M$, in $S$ but as well in $M$. Hence you get a bicollar in $X$. You obtain good candidates for Mayer Vietoris. This will yield the solution. You may want to use Lefschetz Duality and also look closely at the inclusion maps which induce $H_1(\partial M) \to H_1(M) \oplus H_1(S)$. Note also that this is not a trivial Mayer Vietoris case.

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