Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha\_1,\ldots,\alpha\_n,\beta\_1,\ldots,\beta\_n\}$, which are disjoint except for $\alpha\_i$ and $\beta\_i$ intersecting in a single point (relative orientations chosen in the appropriate canonical way).


The $\alpha$'s and $\beta$'s generate $\mathbb{Z}^n$ subgroups $A,B$ of $H\_1(X)$, such that $H\_1(X)=A+B$, and the intersection form restricts to zero on each subspace.


But does every decomposition of $H\_1(X)$ into such a pair of subspaces arise in this way?


(This is very closely related to my question on Maths.SE [here](https://math.stackexchange.com/questions/803015/homology-subgroups-generated-by-non-intersecting-cycles))



Does every canonical decomposition of the intersection form come from a canonical homology basis?
Yes, every such decomposition gives rise to a symplectic automorphism $h$ of $Z^{2n}$ (sending standard symplectic generators to the generators of $A$ and $B$ respectively). 
Now, use the fact that the natural homomorphism from the mapping class group of $X$ to $Sp(2n,Z)$ is surjective. 


