Ref: "Standard" Action of S^1 on S^n ?

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Discussion Overview

The discussion revolves around the concept of the "Standard" action of the circle group ##S^1## on the n-sphere ##S^n##, exploring its definitions, implications, and related properties. Participants examine specific cases, particularly the action on ##S^3## and its relation to projective spaces, while also considering the fixed point property in odd-dimensional real projective spaces.

Discussion Character

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

Main Points Raised

  • One participant seeks a general definition of the action of ##S^1## on ##S^n##, noting specific actions on ##S^3##.
  • Another participant expresses unfamiliarity with a standard action of the circle on spheres and requests references.
  • A participant mentions the use of this action as a counterexample to demonstrate that ##\mathbb{RP}^n##, for odd n, does not possess the fixed point property.
  • It is proposed that ##S^1## acts on ##S^n## by multiplying tuples of complex numbers, which preserves antipodal points.
  • Some participants discuss the implications of the action on projective space, noting that certain actions do not yield fixed point-free maps.
  • There is a discussion about the nature of the fixed point property and its implications for continuous self-maps on projective spaces.
  • One participant suggests that the action of ##S^1## on ##S^n## could be seen as arising from collapsing a cylinder, leading to a natural action on each slice.

Areas of Agreement / Disagreement

Participants express differing views on the existence and nature of a standard action of ##S^1## on ##S^n##, with some proposing specific actions and others questioning their implications, particularly regarding fixed points in projective spaces. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight the complexity of the definitions and the implications of the actions discussed, particularly in relation to fixed points and projective spaces. There are references to specific mathematical properties and assumptions that are not fully explored or agreed upon.

  • #61
That is very interesting and I have not grasped it fully yet. I wonder if I misunderstood something. Rather than the choice of multiplication by e^it or e^-it, it seems that Artin's action does both, i.e. infrareds action has two multiplications, and Artin seems to use e^it as one multiplication but e^-it as then other. So is it a mixture of the two cases you discuss?

I.e. Artin's apparently sends (a,b) to (a.e^it, be^-it). I assume this just by multiplying the matrices he gives on pages 272 and 275, in equations (2.4) and (2.13). I must be misunderstanding something.

well i just noticed it depends on whether i let my matrix act from the left or fom the right! one sends (a,b) to (a.e^it, b.e^it) and the other gives what I wrote 3 lines above. Obviously I need to do some calculations and stop just speculating.
 
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  • #62
mathwonk said:
@WWGD: I added an explicit link now, in post #41, that (may) take a lot less time than reading the book, or not. The point is if you know the (total) chern classes of projective space, then you can compute the chern classes of hypersurfaces in projective space too.

If you have the inclination, you might like my notes on RRT there. I have enjoyed studying it all my career and tried to lay out all I learned there, admittedly not everything there is. Well honestly I only studied the curve case mostly all those years, but did enjoy working up these notes on the surface case and 3-fold case.
Seems link is dead, leads to a forbidden error 403
1664320216389.png
 
  • #65
my pleasure. it seems our webpages get rearranged occasionally, i.e. every ten years or so.
 
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