Questions on Orbifolds S^1/Z_2 and T^2/Z_2

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In summary, the conversation discusses two spaces S^1/Z_2 and T^2/Z_2 and the confusion surrounding their construction and interpretation. The first space is interpreted as [0,1] with the usual action of Z_2, while the second space is the two-sphere but its geometric construction is unclear. The speaker also mentions being familiar with quotient spaces in abstract algebra but unsure about how Z_2 plays a role in these orbifolds. They request further clarification on this topic.
  • #1
"pi"mp
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Hi all, I have a few questions on the two spaces S^1/Z_2 and T^2/Z_2. Am I correct in saying that the first space S^1/Z_2 is simply [0,1] where we interpret Z_2 as being the usual action x~-x? Likewise, I know T^2/Z_2 is simply the two-sphere but I can't quite imagine how this is constructed geometrically. Any hints?

Finally, I have a good grasp on quotient spaces from abstract algebra. However, I'm confused by these orbifolds seeming to tell us to "mod out by Z_2" when Z_2 isn't a subgroup of S^1 or T^2. I think it has something to do with group actions but can someone please elaborate on this for me?

Much appreciated!
 
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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 

1. What is an orbifold?

An orbifold is a generalization of a manifold, which is a mathematical space that is locally similar to Euclidean space. An orbifold allows for certain types of singularities or "orbifold points" where the space is not smooth. It can be thought of as a space that is locally a quotient of a manifold by a finite group of symmetries.

2. What is the significance of the S^1/Z_2 orbifold?

The S^1/Z_2 orbifold is significant in both mathematics and physics. It is a one-dimensional orbifold (a circle) with a Z_2 group acting on it, resulting in a "folded" or "reflected" circle. In mathematics, it is used to study group actions and symmetry breaking. In physics, it is used to describe certain types of spacetime geometries, such as in string theory.

3. How is the T^2/Z_2 orbifold different from the S^1/Z_2 orbifold?

The T^2/Z_2 orbifold is a two-dimensional orbifold (a torus) with a Z_2 group acting on it. It is different from the S^1/Z_2 orbifold in that it has two dimensions instead of one. This results in a more complex geometry and more possible symmetries.

4. What is the relationship between orbifolds and quotient spaces?

Orbifolds can be thought of as a generalization of quotient spaces. In a quotient space, a group acts on a space and the resulting space is made up of equivalence classes of points under the action of the group. In an orbifold, there can be singular points where the space is not smooth. These singular points correspond to the equivalence classes in the quotient space.

5. Can orbifolds be used to describe physical phenomena?

Yes, orbifolds have applications in physics, specifically in string theory and related fields. They can be used to describe certain types of spacetime geometries and symmetries, making them useful tools in understanding the fundamental laws of the universe.

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