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Mathematics
Differential Geometry
Symplectic Basis on Sg and Non-Trivial Curves
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[QUOTE="Bacle, post: 3419368, member: 226962"] Hi, All: Let Sg be the genus-g orientable surface (connected sum of g tori), and consider a symplectic basis B= {x1,y1,x2,y2,..,x2g,y2g} for H_1(Sg,Z), i.e., a basis such that I(xi,yj)=1 if i=j, and 0 otherwise, where I( , ) is the algebraic intersection of (xi,yj), e.g., we may take xi to be meridians and yj to be parallel curves. Does it follow that every non-trivial (non-bounding) SCCurve in Sg 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. [/QUOTE]
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Symplectic Basis on Sg and Non-Trivial Curves
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