How does the intersection form change when changing coefficient rings?

[Bacle]

Hi, All:

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?

 

[mathwonk]

well there is a theory of mod 2 intersection. obviously it is different since it measures only the parity of the number of intersections.
see milnor's topology from the differentiable viewpoint, or guillemin pollack's differential topology.
what does your question mean?

 

[Bacle]

I mean that we go from homology over Z-integers to homology over Z/2 by doing mod-2 reduction, using universal coeff. theorem, etc.

So, say we evaluate the intersection of 2 (transversely-intersecting) classes a,b in H_1(M,Z). We then do a change of coefficients to Z/2 , and so under this change of coefficients, a is sent to a' , b is sent to b'. Is the intersection number (a,b) the same as the intersection number (a',b')?

 

[mathwonk]

obviously an integer cannot equal an integer mod 2, so i suppose you mean does the mod 2 intersection number equal the integral intersection number mod 2? of course the answer is yes. did you consult any of the references i gave?

 

[zhentil]

I mean that we go from homology over Z-integers to homology over Z/2 by doing mod-2 reduction, using universal coeff. theorem, etc.

So, say we evaluate the intersection of 2 (transversely-intersecting) classes a,b in H_1(M,Z). We then do a change of coefficients to Z/2 , and so under this change of coefficients, a is sent to a' , b is sent to b'. Is the intersection number (a,b) the same as the intersection number (a',b')?

You have to be more specific. You're asking if some square is a commutative diagram, but I only know what two of the vertices are. Are you defining the intersection product on H_n(M;Z) or the free part of that? Are you defining the mod 2 intersection pairing on the image of the free part of H_n(M;Z), or on the whole thing?

If you define the intersection pairing on the free part of H_n(M;Z) and you define the mod 2 pairing on the image of that mod 2, then the answer is yes, basically by definition. But if you define it on the whole thing, I believe the answer is no in general.