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lavinia

Science Advisor

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## Main Question or Discussion Point

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?

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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