I've been thinking...and am starting to think that I don't understand complex projective space...So, it's defined as ( Cn+1 \{0,0} / C\{0} ). Now, I think this is just the set of planes in 4 space that pass through the origin... and one can consider how they would all intersect a 3 sphere and think of it as S3/U(1) where U(1) is the circle group... and the hopf function will take all these circles and map them to the 2 sphere isomorphically... but the problem I have is... just pick any 3 of the 4 basis vectors in C^2 and span two planes with them...essential you can just look at R^3 for this... and think of the plane spanned by XY and XZ....well they intersect at the whole X axis...which means there are elements that belong to both planes...but in the case of ( Cn+1 \{0,0} / C\{0} ), these planes are supposed to be equivalence classes...meaning it should divide the space into disjoint sets...and thus, you can't have an element in 2 equivalence classes...Unless, both these planes are actually in the same equivalence class, which is just mind blowing since you can find a bunch more planes that will intercet XY and XZ and before you know it, all of R^3 will belong to the same equivalence class...So, clearly something is wrong with this way of thinking of it...Anyone?(adsbygoogle = window.adsbygoogle || []).push({});

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# Complex Projective Space

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