(This is exercise 2.2.28 from Hatcher) Consider the space obtained from a torus $T^2$, by attaching a Mobius band $M$ via a homeomorphism from the boundary circle of the Mobius band to the circle $S^1\times \{x_0\} \subset T^2$.
We set $A=T^2$, $B=M$, $X:=A\cup B $, $A\cap B\simeq S^1$, so the (reduced) Mayer-Vietoris sequence yields
$$ 0\to \tilde H_2(T^2)\oplus\tilde H_2(M)\to \tilde H_2(X)\to \tilde H_1(S^1)\to \tilde H_1(T^2)\oplus \tilde H_1(M)\to \tilde H_1(X) \to 0$$ and replacing the homology groups we already know we have $$0\to \mathbb Z\overset{\alpha}{\to} \tilde H_2(X) \overset{\beta}{\to} \mathbb Z \overset{\gamma}{\to} \mathbb Z^2\oplus \mathbb Z\overset{\delta}{\to} \tilde H_1(X)\to 0$$
Now, I have to figure out what the maps $\alpha,\beta,\gamma,\delta$ do, and this is where I'm stuck. I'd appreciate a detailed explanation.