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The intersection form ( , ): H_n(M,R)xH_n(M,R)-->Z ; Z the integers and R any coefficient ring, in a 2n-manifold is well-defined in homology, i.e.,

if (x,y)= c , and x~x' and y~y' , then (x',y')=c

Still, how is the value of the intersection form affected by changes in the coefficient ring R? Specifically: what if R went from being torsion-free, like, say, the integers, to having torsion. What would be the difference?

What makes me think that there actually is a difference is that the symplectic groups

Sp^2(2g,Z) and Sp(2g,Z) , which are respectively:

i) Sp^2(2g,Z): The automorphisms of H_1(Sg,Z/2) that preserve intersection, and

ii) H_1(Sg,Z) : automorphisms of H_1(Sg,Z) that preserve intersection

are different groups (actually, I think i) is a subgroup of ii )

Any ideas?

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# Intersection and Coefficients

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