All:
Let $S_g$ be the genus-g orientable surface (connected sum of g tori), and consider
a symplectic basis B= {$x_1,y_1,x_2,y_2,..,x_{2g},y_{2g}$} for $H_1(Sg,\mathbb Z)$, i.e., a basis such that
$I(x_i,y_j)=1$ if i=j, and 0 otherwise, where I( , ) is the algebraic intersection of $(x_i,y_j)$,
e.g., we may take $x_i$ to be meridians and $y_j$ to be parallel curves within the same sub-torus. Does it follow
that every non-trivial (non-bounding) SCCurve in $S_g$ must intersect one of the
curves in B? I think the answer is yes, since, algebraically, every non-bounding curve
is a linear combination of elements in B. Is this correct? Can anyone think of a more
geometric proof?
Thanks.