I am trying to learn a little bit of geometric group theory from the online lecture notes by Cornelia Drutu and Michael Kapovich titled geometric group theory. Here is the link https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf. I am stuck trying to understand the following. I am sorry that i cannot establish the notations and terminology here. Look at page 37 first paragraph of the notes.
Let $G$ be a group with set of generators $X$ and with set of relations $R$ and $A$ any abelian group. Let $Y^2$ be the associated presentation complex. Now we embed $Y^2$ into a $3-$connected cell complex $Y$. It is claimed that $H^2(Y) = H^2(G,A)$.
I have the following questions:
(i) What is meant by attaching appropriate 3-cells. Does that in some way depend upon the abelian group A.
(ii) When computing cohomology of $Y$, how does $A$ comes into the picture.
Any help would be appreciated. Thanks.