- #26

mathwonk

Science Advisor

Homework Helper

- 10,962

- 1,137

This is explained in the post #23 of zinq. I.e.a closed ball in C^2 contains a topological copy of C^2, i.e. the interior of the ball, but with a boundary which is a 3 sphere, whereas P^2 is a copy of C^2 plus a "boundary" P^1 which is a 2 sphere. Thus there is a continuous map from the closed ball to P^2, mapping the 3 sphere boundary of the ball, to the 2 sphere "boundary", or rather the 2 sphere "at infinity", of P^2, by the Hopf map S^3-->S^2. This is not a bijective map, since it drops dimension. The fibers of the Hopf map are circles. This is called "attaching" a 4 cell to the 2 sphere. In general (complex) P^n, a real 2n dimensional manifold, only has cells in even (real) dimensions up to 2n, i.e. it has cells in real dimensions 2,4,....,2n, which are attached by the construction zinq described, and summarized in his last sentence. If you have seen the concept of topological homology, complex P^n has zero homology groups in odd dimensions, and one dimensional homology groups in (non negative) even dimensions up to 2n. Nonetheless, the continuous surjection from the compact closed ball in C^2, onto P^2 does prove P^2 is also compact.

Indeed we cannot have P^2 homeomorphic to a closed ball, since then it would not be a manifold, but would be a "manifold with boundary". The P^1 which is added at infinity to C^2 to make P^2, is not a boundary in the sense of manifold theory, since it has the wrong dimension. The representation of P^2 just given, as the image of a closed ball does not make it obvious that P^2 is actually a manifold, i.e. everywhere locally Euclidean, but that can be seen from the more symmetric definition of P^2 as complex lines through the origin of C^3. I.e. the P^1 at infinity can be taken to be the lines contained in any choice of fixed plane through the origin.

E.g. with coordinates x,y,z on C^3, the line at infinity can be chosen as lines in either the x,y plane or the y,z plane, or the z,x plane. The 3 complements of those 3 choices, gives a cover of P^2 by three different copies of C^2. E.g. the set of complex lines through the origin but

Indeed we cannot have P^2 homeomorphic to a closed ball, since then it would not be a manifold, but would be a "manifold with boundary". The P^1 which is added at infinity to C^2 to make P^2, is not a boundary in the sense of manifold theory, since it has the wrong dimension. The representation of P^2 just given, as the image of a closed ball does not make it obvious that P^2 is actually a manifold, i.e. everywhere locally Euclidean, but that can be seen from the more symmetric definition of P^2 as complex lines through the origin of C^3. I.e. the P^1 at infinity can be taken to be the lines contained in any choice of fixed plane through the origin.

E.g. with coordinates x,y,z on C^3, the line at infinity can be chosen as lines in either the x,y plane or the y,z plane, or the z,x plane. The 3 complements of those 3 choices, gives a cover of P^2 by three different copies of C^2. E.g. the set of complex lines through the origin but

*not*lying in the x,y (i.e. z=0) plane, is bijectively equivalent to the set of points of the plane z=1, which is a copy of C^2. I.e. each complex line through the origin but not lying in z=0, contains exactly one point whose z coordinate is 1.
Last edited: