What is S^1/Z_2 and its connection to the projective plane?

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SUMMARY

S^1/Z_2 represents the projective plane, denoted as \textbf{P}^2. The notation Z_2 refers to the additive group of integers modulo 2, consisting of the elements {0, 1}. In this context, S^1 is the unit circle, and modding by Z_2 identifies each point on the circle with its diametrically opposite point. This results in a space that can be visualized as the upper half-circle where the endpoints (0 degrees and 180 degrees) are identified.

PREREQUISITES
  • Understanding of basic topology concepts, specifically S^1 (the unit circle).
  • Familiarity with group theory, particularly Z_2 (integers modulo 2).
  • Knowledge of projective geometry and the properties of the projective plane.
  • Basic mathematical notation and terminology used in topology and geometry.
NEXT STEPS
  • Study the properties of projective spaces, focusing on \textbf{P}^2 and its applications.
  • Explore the concept of quotient spaces in topology to understand the process of modding by groups.
  • Learn about the implications of identifying points in topological spaces and how it affects their properties.
  • Investigate the relationship between S^1/Z_2 and other geometric constructs, such as the Klein bottle.
USEFUL FOR

Mathematicians, students of topology, and anyone interested in understanding the connections between different geometric spaces, particularly in the context of projective geometry and higher-dimensional theories.

touqra
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I encountered in a paper this space: S^1/Z_2
What kind of space is this? What is Z_2 ?

Thought I should ask you guys, cause the paper was about extra dimension stuffs.
 
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touqra said:
I encountered in a paper this space: S^1/Z_2
What kind of space is this? What is Z_2 ?

Thought I should ask you guys, cause the paper was about extra dimension stuffs.

Really should ask it in math forum

Z_2 is the additive numbers mod 2, that is the set of numbers {0,1}
where when you add you chop out any two, so 1+1 = 0

S^1 is the circle
so when you mod you just get all the LINES thru the origin, in the plane.

so you could think of the set you said as roughly equal the upper half-circle

or as all the angles from 0 to 180 degrees, but with 0 degrees and 180 degrees identified (stuck together)

modding by Z_2 just identifies each direction with MINUS that direction
 
Last edited:
And S^1/Z_2 is the space you get when identifying each point on the circle with its opposite point. It's \textbf{P}^2, the projective plane.
 

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