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)


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.


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