When playing around with a homological calculation I came across a short exact sequence of the form $ 0 \to \mathbb Z^{2g} \to H \to \mathbb Z / 2 \to 0 $ My background in algebra is not very strong. Does this sequence already imply the form of $H$?
For a lot of the calculations I am unsure when to switch from geometry to algebra and vice versa, i.e. how much geometrical information about the maps is necessary to pin down the homology group. It would be great if you could point me to some references where examples are worked out in detail.