A question about full pure mapping

Question

Does there exist a full pure mapping from Unit Circles to Complex Plane

Any hint will be appreciated

Answer 1

I will be honest. I am not a mathematician. However, consider the unit circle centered at (x,y,z) = (0,0,1). Form the unit sphere centered at the same point. The question becomes, "Is there a surjective map from the unit circle (the equator of the unit sphere) onto the complex plane containing (0,0,0)?"

I don't know, do you? Allowing the compactification of unit sphere stereographically projected onto the entire complex plane by adding the point at infinity is a common placebo offered to cover the infinite potential of the North Pole.

However, your question is about mappings of unit circles onto the complex plane. The Great Circles of the unit sphere passing through the North Pole are unit circles. Allowing the North Pole to be mapped onto the point at infinity, perhaps it is clear every stereographic projection of such Unit Circles gets mapped surjectively onto the complex plane.

I am a mathematical economist, knowing little of mathematics. Perhaps this and ten cents will buy you a cup of coffee (pre-inflation prices). Sorry, I truly hope this helps.