How many fixed points on a circle S/Z2?

In summary, the question asks to show that there are two points on the circle that are left fixed by the Z2 action, which identifies x ~ x + 2 on the space -1 < x <= +1. It is known that x = 0 is one of the fixed points, and after some discussion, it is determined that x = 1 is the other fixed point. It is noted that this result holds because the Z2 mod and the identification on the fundamental domain both have the same invariant sets. The concept of fixed points and invariant sets is clarified.
  • #1
Living_Dog
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0
I am working on Zweibach's First Course in String Theory and question 2.4 asks: Show that there are two points on the circle that are left fixed by the Z2 action. (For those without the text, the circle is the space -1 < x <= +1, identified by x ~ x + 2. And the Z2 mod imposes the x ~ -x identification on the circle.)

I know that x = 0 is one of the fixed points, but the other alludes me. Just a guess, is it the center? but that is not in the fundamental domain!


...eek!


Thanks in advance,
-LD
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  • #2
Hi Living_Dog!

x = 1 is fixed.

It goes to x = -1, and x = x + 2, so -1 = 1. :smile:
 
  • #3
tiny-tim said:
Hi Living_Dog!

x = 1 is fixed.

It goes to x = -1, and x = x + 2, so -1 = 1. :smile:

After thinking about it I think you are saying that:

x ~ -x so 1 goes to -1 for the Z2 mod.

THEN

x ~ x + 2 takes the -1 and it goes to -1 + 2 = +1 = itself.

Great. I see that now, but is this the general approach? First the mod and then the identification on the fundamental domain (f.d.)?

Let's take the f.d. identification 1st, namely:

x ~ x + 2 takes +1 to 3, which on the space of the circle, -1 < x <= +1 is 1 ... hmmm. This goes to itself already. Why include the Z2 mod?

I'm sorry I am asking such basic questions but I only had 1 topology course a million years ago, no group theory, and ... am not that bright to begin with! :blushing: (Tomorrow I am getting Introduction to Compact Transformation Groups by Berdon. Hopefully that will help.)Thanks,
-LD
_______________________________________
my bread: http://www.joesbread.com/
my faith: http://www.angelfire.com/ny5/jbc33/
 
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  • #4
Living_Dog said:
After thinking about it I think you are saying …

:smile: … sorry to make you think! … :smile:

x ~ x + 2 takes +1 to 3, which on the space of the circle, -1 < x <= +1 is 1 ... hmmm. This goes to itself already. Why include the Z2 mod?[/SIZE])

The two mods just happen to have the same invariant sets.

But if you defined, for example, x ~ -x + .1, then they wouldn't!

is this the general approach? First the mod and then the identification on the fundamental domain (f.d.)

I think you can do them in either order …

btw, I don't know what the examiners' view on this is, but the reason I've been writing "=" instead of "~" is that, once you've used ~ to create the space, the "two points" are one point! :smile:
 
  • #5
tiny-tim said:
:smile: … sorry to make you think! … :smile:

The two mods just happen to have the same invariant sets.

You mean since -1 < x <= 1 has the same range as 0 < x <= 2?

tiny-tim said:
But if you defined, for example, x ~ -x + .1, then they wouldn't!

I think you can do them in either order …

btw, I don't know what the examiners' view on this is, but the reason I've been writing "=" instead of "~" is that, once you've used ~ to create the space, the "two points" are one point! :smile:

But why include the Z2 mod if the ... unless one is out to fix two points (requiring both id's) instead of only one.

Now that I think of it, there is no fixed point for the id x ~ x + 2, yes?Thanks for your help. Don't apologize for making me think. I teach a course on critical thinking. :smile:

-LD
_______________________________________
my bread: http://www.joesbread.com/
my faith: http://www.angelfire.com/ny5/jbc33/
 
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  • #6
Living_Dog said:
You mean since -1 < x <= 1 has the same range as 0 < x <= 2?

Sorry - maybe I'm using no-standard terminology - by "invariant", I just meant any fixed point (any point that ~ itself).

Now that I think of it, there is no fixed point for the id x ~ x + 2, yes?

Hurrah! :smile:
 
  • #7
tiny-tim said:
Sorry - maybe I'm using no-standard terminology - by "invariant", I just meant any fixed point (any point that ~ itself).

I see:

fixed = invariant

points = set

tiny-tim said:
Hurrah! :smile:

I always was a fan of mathematics... it seems so useful. :smile:


Thanks!

-LD
________________________________________
my bread: http://www.joesbread.com/
my faith: http://www.angelfire.com/ny5/jbc33/
 
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1. How many fixed points are there on a circle with a Z2 symmetry?

The number of fixed points on a circle with Z2 symmetry is two. This is because the Z2 symmetry group has two elements, the identity element and a single non-trivial element. The non-trivial element inverts the circle, leaving only two points unchanged.

2. Can there be more than two fixed points on a circle with a Z2 symmetry?

No, there cannot be more than two fixed points on a circle with Z2 symmetry. This is because the Z2 symmetry group only has two elements, and each element can only fix one point on the circle.

3. How are the fixed points spaced on the circle with Z2 symmetry?

The fixed points on a circle with Z2 symmetry are spaced evenly apart, with a distance of half the circumference of the circle between them. This is because the non-trivial element in the Z2 symmetry group inverts the circle, effectively flipping the positions of the two fixed points.

4. Is the number of fixed points affected by the size of the circle?

No, the number of fixed points on a circle with Z2 symmetry remains the same regardless of the size of the circle. This is because the Z2 symmetry group and its two elements remain unchanged, regardless of the size of the circle.

5. How do fixed points on a circle with Z2 symmetry relate to other symmetries?

Fixed points on a circle with Z2 symmetry are a special case of fixed points in other symmetries. For example, a circle with Z3 symmetry will have three fixed points, evenly spaced apart. In general, the number of fixed points on a circle will depend on the symmetry group and its elements.

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