Interesection Numbers in RP^3

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lavinia
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The Hopf invariant of a map of the 3 sphere into the 2 sphere can be calculated as the integral over the 3 sphere of W^dV where W and dV are the pull backs of two bump 2 forms with disjoint support centered around two regular values of the map.

This integral can be interpreted as the linking number of the two circles that are inverse images of the two regular values and W and dV are the Poincare duals of these two circles.

A similar calculation can be done on projective 3 space. While not simply connected, its fundamental group is torsion and its second de Rham cohomology group is zero. All of the same arguments used to verify that the Hopf invariant is well defined and is a homotopy invariant go through for maps of projective 3 space into the 2 sphere.

So my question is whether one can interpret the integral of W^dV as a linking number in projective space. I don't think so but it must be close and perhaps suggests a generalization of the idea of linking number.

Here is an example calculation. This example came from a previous thread where the non-triviality of the tangent circle bundle of the 2 sphere (equivalent to the 2 sphere not having a continuous everywhere non-zero vector field) was proved by noticing that it is homeomorphic to projective 3 space.

At any point on the two sphere choose a geometry where the sphere has its usual shape with Gauss curvature 1 near the point but which rapidly becomes flat (zero Gauss curvature ) outside of a small open ball around the point. This sphere looks like a small dome sitting on a flat disk, kind of a flying saucer shape. The Gauss curvature is constant on most of the dome but is zero outside of the ball and so is close to a bump function.

As usual the exterior derivative of the connection 1 form, dV, equals -K times the pull back of the volume element, where K is the Gauss curvature of the associated metric. Dividing this form by -1/4pi produces the Poincare dual of the fiber circle above the top point of the dome on the sphere. This normalized form is d(-1/4pi)V.

If W is (-1/4pi) times the 2 form for the same geometry around another point on the sphere then the integral of the wedge product is (8pi)/(16 pi^2) or 1/2. this seems to make sense because the 3 sphere is a two fold cover of projective 3 space.Of course 1/2 is not a linking number but somehow the construction seems to be the same.

Any help?
 
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mathwonk
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do you have a definition of linking number in this setting? or are you just saying that this construction behaves as if it should be a generalized one? I guess you are implying the usual definition as an intersection number is not quite apt since one cannot cap off the curves uniquely in a non simply connected space.
I recall Dold saying the linking number of two loops is the intersection number of one loop with the disc capping off the other one. So maybe here you have more than one choice of disc and you are averaging? the answers somehow?

you might try this out on mathoverflow. All I know is this memory from sitting in that one day in Dold's class 45 years ago.
 
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lavinia
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do you have a definition of linking number in this setting? or are you just saying that this construction behaves as if it should be a generalized one? I guess you are implying the usual definition as an intersection number is not quite apt since one cannot cap off the curves uniquely in a non simply connected space.
I recall Dold saying the linking number of two loops is the intersection number of one loop with the disc capping off the other one. So maybe here you have more than one choice of disc and you are averaging? the answers somehow?

you might try this out on mathoverflow. All I know is this memory from sitting in that one day in Dold's class 45 years ago.
I think it might slightly generalize linking numer. In Bott and Tu they compute linking numbers of fibers above regular points of a map from the 3 sphere into the 2 sphere by the same construction.

I looked at mathoverflow and was a bit daunted. It seems to be at a way higher level than I am at and I wondered if a question like this would be appropriate. There is amazing stuff on it. thanks for pointing it out to me.
 
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lavinia
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I think it might slightly generalize linking numer. In Bott and Tu they compute linking numbers of fibers above regular points of a map from the 3 sphere into the 2 sphere by the same construction.
The thought that seems right is that the tangent circle bundle is the Hopf fibration modulo the antipodal map on S^3. If this is right then these form should pull back to the forms on S^3 used in calculating the Hopf invariant.

The idea for a proof - S^3 acts on R^3 by rotations. This action can be realized as the action of the unit quaternions on the purely imaginary quaternions by quaternionic conjugation. Antipodal quaternions produce the same rotation so this action projects to projective 3 space.

The stabilizer of a point on the 2 sphere is a Hopf circle in the unit quaternions. This circle is naturally identified with the tangent circle above the stabilzed point modulo antipodal points. This means that the Hopf fibration is factored by a two fold covering through the tangent circle bundle of the sphere.

Therefore, the linked circles of the Hopf fibration project to two tangent circles on the 2 sphere.

The problem then is to visualize these two tangent circles and see how they are linked.
 
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mathwonk
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My remark was flawed also. I.e. the fact you space is not simply connected argues with the existence, not the uniqueness of the disc capping off a loop. I.e. if every loop were capped off by some disc, we would have ∏1 =0. But the fact that the second cohomology is zero, wait a minute de rham cohomology? that ignores torsion doesn't it? oh but so does intersection number i guess.

anyway, according to Dold, to define a linking number, you cap off one loop with a disc and then intersect the other loop with that disc. so in a space with ∏1 = Z/2, I guess you might need to take a disc that winds twice around the loop to cap it off, and then intersect with that,. but then you should probably divide by 2?

these differential form constructions are nice but do not always have the intuition of the original meaning of a term like linking number.

I think your question will be fine on mathoverflow, and if not, they will say so nicely and suggest you try it on stackexchange. if you do not want to take a chanc, try stackexchange.
 

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