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

[touqra]

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.

 

[marcus]

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

 

[CompuChip]

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.