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Spinnor

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Suppose I am stuck inside a three-sphere, S^3, given by w^2 + x^2 + y^2 + z^2 = 1. Let me be near the surface where w is zero. Let the surface w = 0 be painted white, this surface divides the three-sphere into two parts? If so say I am stuck on one side.

Is it true that at each point of this painted surface, w = 0, I can attach a little vector which would represent each fiber of a Hopf Fibration for my three-sphere above where the painted two-sphere is to be fibered? I would guess this is not unique? Is there a simple function that would give the direction of my little vectors on the surface w = 0, I want to imagine walking on one side of my surface w = 0 and picturing the little arrows? Hope this is clear and makes sense.

Edit, If the fibers are oriented then the vectors above would come in pairs for each fiber, one pointing "up" and the other pointing "down"? A single fiber both enters and exits "my side" of the three-sphere?

Thanks for any help.

Is it true that at each point of this painted surface, w = 0, I can attach a little vector which would represent each fiber of a Hopf Fibration for my three-sphere above where the painted two-sphere is to be fibered? I would guess this is not unique? Is there a simple function that would give the direction of my little vectors on the surface w = 0, I want to imagine walking on one side of my surface w = 0 and picturing the little arrows? Hope this is clear and makes sense.

Edit, If the fibers are oriented then the vectors above would come in pairs for each fiber, one pointing "up" and the other pointing "down"? A single fiber both enters and exits "my side" of the three-sphere?

Thanks for any help.

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