variety
Nov11-09, 08:21 PM
1. The problem statement, all variables and given/known data
This isn't really a homework problem, but whatever. So in my textbook, it says that "of course S^1/Z_2 is isomorphic to S^1." I want to know why.
2. Relevant equations
3. The attempt at a solution
Is S^1 even a group? Well I guess it can be a group, with multiplication as the operation (S^1 viewed as a subset of C). But then the only map I can think of between the two is the projection p:S^1\rightarrow S^1/Z_2, where x\in S^1 is mapped to its obit under the action of Z_2. But this map is clearly not injective since p(x)=p(-x) for any x \in S^1.
If this is a typo in my book, then are the sets homeomorphic (maybe they wanted to use the symbol for homeomorphism instead of isomorphism)? What then is the map between them?
This isn't really a homework problem, but whatever. So in my textbook, it says that "of course S^1/Z_2 is isomorphic to S^1." I want to know why.
2. Relevant equations
3. The attempt at a solution
Is S^1 even a group? Well I guess it can be a group, with multiplication as the operation (S^1 viewed as a subset of C). But then the only map I can think of between the two is the projection p:S^1\rightarrow S^1/Z_2, where x\in S^1 is mapped to its obit under the action of Z_2. But this map is clearly not injective since p(x)=p(-x) for any x \in S^1.
If this is a typo in my book, then are the sets homeomorphic (maybe they wanted to use the symbol for homeomorphism instead of isomorphism)? What then is the map between them?