Homology subgroups generated by non-intersecting cycles Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by simple closed curves, all disjoint except for each pair $a_i$ and $b_i$ intersecting at 1 point.
However, there are many such choices of basis. I'd like to know whether there is an algebraic way of classifying all the choices. The concrete question I'm most interested in is whether we can classify the choice of A-cycles, or:
Given a $\mathbb{Z}^g$ subgroup of $H_1$ (expressed in a particular canonical basis), how can one tell if it can be generated by $g$ homologically independent disjoint simple closed curves? Can I classify all such subgroups?
 A: EDIT : Just to make sure that this is clear, this answers the more difficult question asked in the comments.  It also gives a complete answer to the original question.

In general, this is a subtle question and I doubt there is a simple-to-state answer.  See the paper
Edmonds, Allan L.
Systems of curves on a closed orientable surface. 
Enseign. Math. (2) 42 (1996), no. 3-4, 311–339.
for a discussion of some of the difficulties.
But one answer to a restricted version of your question is given by the following fact.  Fix a closed orientable surface $S$.  Say that a set $\{\vec{v}_1,\ldots,\vec{v}_k\}$ of elements of $H_1(S;\mathbb{Z})$ is isotropic if $i(\vec{v}_i,\vec{v}_j)=0$ for all $1 \leq i,j \leq k$ and unimodular if the $v_i$ form a basis for a $k$-dimensional direct summand of $H_1(S;\mathbb{Z})$.
Prop : Let $\{\vec{v}_1,\ldots,\vec{v}_k\}$ be a set of elements of $H_1(S;\mathbb{Z})$.  Then there exist disjoint simple closed orientable curves $\{\gamma_1,\ldots,\gamma_k\}$ with $[\gamma_i] = \vec{v}_i$ for $1 \leq i \leq k$ such that $S \setminus (\gamma_1 \cup \cdots \cup \gamma_k)$ is connected if and only if the set $\{\vec{v}_1,\ldots,\vec{v}_k\}$ is isotropic and unimodular.
For a proof of this, see Lemma 6.2 of my book-in-progress "Lectures on the Torelli group", available here.  You also might find the discussion of realizing symplectic bases in Chapter 2 enlightening.
