How many fixed points on a circle S/Z2?

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Discussion Overview

The discussion revolves around the identification of fixed points on a circle under the action of the group Z2, as posed in a question from a string theory textbook. Participants explore the implications of the Z2 action, which identifies points on the circle in specific ways, and the resulting fixed points that remain unchanged under these transformations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant identifies x = 0 as a fixed point and speculates about the existence of another fixed point, questioning if it could be the center.
  • Another participant asserts that x = 1 is also a fixed point, explaining that it maps to -1 and then back to itself through the Z2 action.
  • There is a discussion about the order of applying the Z2 mod and the identification on the fundamental domain, with some suggesting that the order may not matter.
  • Concerns are raised about the necessity of including the Z2 mod if the identification x ~ x + 2 does not yield any fixed points.
  • Participants discuss the concept of invariant points and how they relate to fixed points, with some confusion about terminology and definitions.
  • One participant expresses uncertainty about the implications of the Z2 mod and its role in fixing points, questioning if both identifications are needed to establish two fixed points.

Areas of Agreement / Disagreement

Participants generally agree that x = 0 and x = 1 are fixed points, but there is uncertainty regarding the necessity and implications of the Z2 mod and the identification x ~ x + 2. The discussion remains unresolved regarding the broader implications of these transformations and their order.

Contextual Notes

There are limitations in the discussion regarding the definitions of fixed points and invariants, as well as the assumptions made about the order of operations in applying the Z2 mod and the fundamental domain identification. Some mathematical steps and reasoning remain unclear.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of string theory, topology, and group theory, particularly those interested in the implications of group actions on geometric spaces.

Living_Dog
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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
_______________________________________
my bread: http://www.joesbread.com/
my faith: http://www.angelfire.com/ny5/jbc33/
 
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Hi Living_Dog!

x = 1 is fixed.

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


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:
 
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/
 
Last edited by a moderator:
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:
 
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/
 
Last edited by a moderator:

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