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

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SUMMARY

The discussion centers on the mathematical concepts of orbifolds S^1/Z_2 and T^2/Z_2. It is established that S^1/Z_2 can be represented as the interval [0,1] with Z_2 acting by the relation x ~ -x. Additionally, T^2/Z_2 is identified as the two-sphere, though the geometric construction remains unclear to some participants. The confusion arises regarding the application of Z_2 as a group action on these spaces, particularly since Z_2 is not a subgroup of S^1 or T^2.

PREREQUISITES
  • Understanding of quotient spaces in topology
  • Familiarity with group actions in abstract algebra
  • Basic knowledge of orbifolds and their properties
  • Concept of the two-sphere and its geometric representation
NEXT STEPS
  • Study the geometric construction of T^2/Z_2 and its implications
  • Explore the concept of group actions in more depth, particularly in relation to topological spaces
  • Investigate the properties and applications of orbifolds in topology
  • Learn about the relationship between quotient spaces and group theory
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This discussion is beneficial for mathematicians, particularly those specializing in topology and abstract algebra, as well as students seeking to deepen their understanding of orbifolds and group actions.

"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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