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Intersection form. Please help with notation

  1. Jan 19, 2008 #1

    WWGD

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    Hi, everyone:

    I am trying to understand the intersection form, and I am having trouble
    with the notation used in Wikipedia's entry on intersection theory:


    http://en.wikipedia.org/wiki/Intersection_theory_(mathematics)

    Now, I am somewhat weak in my cohomology, and I understand the concept
    is necessarily involved, but I would appreciate some comments/insights on
    this:


    Specifically, in the definition of the bilinear form Qm, on the n-th cohomology
    ring H^n(M) :


    Qm: H^n(M,delM,Z) x H^n(M, delM; Z)-->Z , given by:

    Qm(a,b)=< a\/b, [M] >

    where '\/' is the cup product, 'del' is the boundary, and 'Z' are the integers.

    Now:
    What is this bracket operation < , > ?. It is not stated that M is Riemannian

    so I don't see how this would be an inner product. And, what is [M] here ?

    I have been reading up in H&Y (Hocking and Young) , and they are using

    different notation.


    Thanks for Any Help.
     
  2. jcsd
  3. Jan 21, 2008 #2

    mathwonk

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    you are wasting your time reading that. try one of the introductory works on the subject such as fulton's introductory lectures at george mason university in the regional conference series.

    notation is less important than concepts. that bracket you asked about just means evaluation of a cocycle on a cycle. or if you know some integration theory, it is like integrating an n form over an n manifold. but really that formal $%%#@ is a waste of your time.

    even wikipedia admits in the body of that article that it does not meet their own quality standards.
     
  4. Jan 21, 2008 #3

    WWGD

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    Thanks, Mathwonk. Is there a difference in intersection theory in Alg.
    Geometry and in Alg. Topology?. The article in Wiki seemed to suggest this;
    it may not be of good quality, but it is all I know about the topic at the moment.
     
  5. Jan 27, 2008 #4

    mathwonk

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    intersection theory is easy when the two things being intersected are transverse, i.e. meet only where their tangent spaces intersect minimally.

    hence the hard part of intersection theory is replacing non transverse objects by transverse ones, i.e. "moving lemmas".

    in topology there is a huge flexibility in moving things smoothly or continuously, and thus moving lemmas are relatively "easy".

    in algebraic geometry one is constrained, especially in the case of abstract fields of coefficients, to use only subvarieties defiend by aklgebraic equations, so smotholy moving them is impossible.

    one thus uses much finer equivalence relations on subvarieties such as Linear equivalence, or rational equivalence, or algebraic equivalence.

    since these relations are much stricter, the set of equivalence classes one is forced to work with can be huge and thus hard to work with.

    even defining the intersection multiplicity is tricky. shafarevich basic algebraic geometry, has a nice treatment of the easiest case, of invariance of intersection numbers for divisors on a smooth variety.

    fulton is the guru of intersection theory. he has written several books on it, first his book on algebraic [plane?] curves, a lovely book which seems out of print. then his george mason CBMS conference lectures i referred to above, then his tome which i myself have barely scratched the surface of, Intersection theory.

    the smooth topological case is well treated in guillemin and pollack's differential topology, an easy read for strong senior math majors.
     
  6. Jan 27, 2008 #5

    mathwonk

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    according to poincare duality, intersection theory on homology cycles is dual to cup products on cohomology cocycles, and cohomology is always technically easier to deal with than homology.

    in calculus intersection theory is dual to integration of wedge products.

    i.e. dual to each smooth closed path on a riemann surface, is a smooth one form such that integrating another path against this one form is the same as intersecting it with the original path.

    fulton even uses the integration sign as a symbol for the intersection operation in his book.

    you might try reading this part of my free notes on my website on the riemann roch theorem for this treatment. i learned this from reading hermann weyl's great book on riemann surfaces.
     
  7. Jan 27, 2008 #6

    mathwonk

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    oh, if you want the purely topological version of intersection theory, then you replace differential forms by cocycles. this is treated well in the somewhat abstract book of dold on algebraic topology.

    algebraic topology is quite abstract because in some cases they are trying to mimic ideas from calculus without using calculus, i.e. no tangent spaces.

    i greatly prefer differential topology, but as outlined in other places here, the results are different at least in some exotic cases.
     
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