Projective Space: CP1, Homeomorphism to 2-Sphere?

In summary, the conversation discusses the complex projective space CP1 and how it can be represented as the action of C on C^2\{0}. This results in equivalence classes of "lines" passing through the origin in C^2, but not including 0. It is mentioned that any vector in complex space, when multiplied by the set of all complex numbers of a given norm, will give rise to a circle. The discussion also touches on the idea that when C acts on C^2, the result is equivalence classes of planes. The ultimate question is whether this construction is homeomorphic to a 2-sphere, to which it is confirmed that CP1 is indeed the 2-sphere and this construction is one way
  • #1
Thorn
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I have a question about complex projective space... specifically CP1 which can be thought of as the action of C on C^2\{0} which gives rise to the equivalence classes of "lines" passing through the origin in C^2 (but not including the 0) Now, any vector in complex space, when multiplied by the set of all complex numbers of a given norm will give rise to a circle...and in a sense, when C (numbers of all norm!) acts of C^2 you simply get equivalence classes of planes... I think anyway..someone correct me if I am wrong...but the ultimate question is.. is this homeomorphic to a 2 sphere?
 
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  • #2
Yes, CP1 is the 2-sphere. This construction of S2 from C2 is on way to do the celibrated Hopf fibration.
 

1. What is projective space?

Projective space, also known as projective geometry, is a mathematical concept that extends the traditional Euclidean geometry to include points at infinity. It is a space where parallel lines intersect at a point and all points are considered equivalent regardless of their position in space.

2. What is CP1 in projective space?

CP1, also known as the complex projective line, is a specific type of projective space that is used in complex geometry. It is a one-dimensional complex manifold that is homeomorphic to the Riemann sphere, a 2-dimensional sphere.

3. How is CP1 homeomorphic to a 2-sphere?

A homeomorphism is a continuous and bijective function between two topological spaces that has a continuous inverse. In the case of CP1 and the 2-sphere, there is a specific mapping that preserves the topological structure of both spaces, making them homeomorphic. This mapping is known as the stereographic projection.

4. What are the applications of projective space?

Projective space has various applications in mathematics, physics, and computer science. It is used in computer graphics for 3D rendering and in computer vision for image transformations. In physics, it is used to describe phenomena in quantum mechanics and relativity. It also has applications in data analysis, optimization, and machine learning.

5. What is the significance of projective space in mathematics?

Projective space is an important concept in mathematics as it provides a way to study and understand geometric objects in a more general and abstract manner. It also allows for the unification of different geometries, such as Euclidean, hyperbolic, and elliptic geometries. Additionally, projective space has connections to other areas of mathematics, such as algebraic geometry and topology, making it a fundamental concept in modern mathematics.

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