Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Intersection and Coefficients

  1. Jul 31, 2011 #1
    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?
  2. jcsd
  3. Aug 17, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper

    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?
  4. Aug 17, 2011 #3
    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')?
  5. Aug 17, 2011 #4


    User Avatar
    Science Advisor
    Homework Helper

    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?
  6. Aug 17, 2011 #5
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook