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Def./Examples of Spin Structure Re Trivial Bundles?

  1. Aug 12, 2011 #1
    Hi, All:

    I am reading a paper in which , if I understood well, a spin structure in a manifold M

    is equivalent to M admitting a trivialization of the tg.bundle over the 1-skeleton of M

    ( I guess M is assumed to be "nice-enough" so that it is a simplicial complex )

    so that this trivialization extends to a trivialization of the 2-skeleton; then , e.g., S^2 (seen as a

    2-simplex, with 1-simplices as S^1's ) would not be spin, since a trivialization over any 1-subskeleton S^1

    of S^2 cannot extend to a trivialization of the bundle over S^2 --because the

    bundle over S^2 is not trivial, . A spin structure (in this layout) is a choice, up to homotopy

    of a (global) trivialization of the bundle over the 1-skeleton that extends to a trivialization over the

    2-skeleton.


    O.K, so far. But now there is a different use of spin structure which is supposedly

    equivalent to the one just given: a 4-manifold is spin if every homology 2-class can be

    repd. by an embedded sphere; ** BUT ** , while then S^2 is not Spin by the first

    def ( bundle over S^2 is not trivial, by , e.g. the Hairy Ball Thm. ), it is spin by

    the second def., since its second homology class is S^2 itself --because S^2 is

    orientable. Am I missing something here somewhere?

    Any Ideas?

    Thanks in Advance.
     
    Last edited: Aug 12, 2011
  2. jcsd
  3. Aug 12, 2011 #2

    Ben Niehoff

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  4. Aug 12, 2011 #3
    I would think that the second condition doesn't apply to S^2 because it's not 4-dimensional :)

    Anyway, you have to be careful here. In dimensions greater than two, SO(n) has fundamental group Z/2. Spin(n) is the unique double cover of this. However, this doesn't go through for two dimensions, since the double cover of the circle is itself. The condition that does go through in both cases is that w_2(TM) vanishes.

    To see why the condition of passing from the one-skeleton to the two-skeleton doesn't work in two dimensions, let's do the obstruction theory. Fix n>2, and fix a trivialization over the one-skeleton, thought of as a map S^1 -> BSO(n). The obstruction to extending the map over the two-skeleton lies in H^2(S^2, pi_1(BSO(n)))=0, while the obstruction to extending the map as a homotopically trivial map lies in H^2(S^2,pi_2(BSO(n)))=H^2(S^2,pi_1(SO(n)))=H^2(S^2,Z/2), which is precisely w_2. I.e. two SO(n) bundles over S^2 are isomorphic iff they have the same second Stiefel-Whitney class. From Cech cohomology, you see right away that this is precisely the obstruction to lifting the bundle to a Spin(n) bundle, i.e. an SO(n) bundle over S^2 is trivial iff its second Stiefel-Whitney class vanishes iff it's Spin. You also glean from this that the obstruction for any manifold being spin is w_2 being trivial, which is the same thing as the classifying map over the two-skeleton being homotopically trivial.

    This all changes if n=2. Here since Spin(2)=S^1, we want to lift the bundle to an S^1 bundle which is compatible with the double cover S^1 -> S^1 (this is still w_2). You'll notice right away that this has nothing to do with the bundle being trivial over the two-skeleton.
     
    Last edited: Aug 12, 2011
  5. Aug 12, 2011 #4
    Thanks, Zhentil:
    As you may have noticed, this is not my area of expertise.

    Still, is there a connection here with being able to represent homology two-classes

    by embedded spheres ( in which case CP^2 would be spin, since H_2(CP^2)=CP^1 , and

    CP^1 ~ S^2 --actually they are diffeomorphic )? But, as Niehoff points out, Wikipedia

    claims that only CP^(2n+1) is spin . I was just reading some article by Kirby and Friedman

    in which , if I had understood well, they were using a certain quadratic Z/2-form

    ( a quadratic form over Z/2) called the Rokhlin form ; defined on H_1(Sg,Z/2) ;(I think

    Sg was an embedded copy of g-torus on the manifold M in question) to determine if a

    manifold was spin. The result was then that the manifold was spin if the form was killed

    on a symplectic basis {x1,y1,x2,y2,...,x2g,y2g} (this is a basis for the 1-homology

    of the embedded g-torus g-torus, in which (xi,yj)=1 if i=j , and (xi,yj)=0 otherwise )

    So, basically, if the Rokhlin form was zero on the symplectic basis, then the handles

    of the 2-class rep. could be surgered away without affecting the overall homology.

    Sorry for running off; the point is it seemed like they were arguing that if the manifoild

    was spin, then the handles in the 2-class could be surgered away, and the class would

    be represented by an embedded sphere.
     
  6. Aug 12, 2011 #5
    No, there is no way that CP^2 is spin. w_2 is the mod 2 reduction of the first Chern class, which is non-zero for CP^2.
     
  7. Aug 12, 2011 #6
    Thanks, Zhentil; I don't mean to be argumentative, I just want to clarify some things:

    so being spin has no relation then with representing homology 2-classes (in a 4-manifold )

    by embedded spheres, i.e., with being able to lower the genus of the representatives

    for H_2(M^4)?

    P.S: thanks for your patience; I am doing way too much stuff and I sometimes skip
    on important details; hopefully that will change soon.
     
  8. Aug 12, 2011 #7
    No, it's certainly possible that that's true. I can't prove it off the top of my head, but I just wanted to point out that CP^2 isn't a counterexample, since the higher multiples of CP^1 aren't representable by embedded spheres.
     
  9. Aug 12, 2011 #8
    Zhentil:Sorry for the bother, just to know what B(SO(n)) stands for. Also wondering if you got some
    good refs. for this topic.
     
    Last edited: Aug 12, 2011
  10. Aug 13, 2011 #9
    It's no trouble at all :) BSO(n) is the classifying space for SO(n) bundles. Other people might be able to help you more on the references, but I learned what I know from Milnor's characteristic classes, Differential Topology by Hirsch, and Differential Forms and Rational Homotopy Theory by Griffiths and Morgan (this one is the best imo for obstruction theory).
     
  11. Aug 13, 2011 #10
    Also Kirby Calculus by Gompf and Stipsicz is an amazing book. It's a good place to learn, e.g., that a spin structure is equivalent to a trivialization over the one-skeleton that extends over the two-skeleton.
     
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