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

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SUMMARY

The discussion centers on the "Standard" action of the circle \( S^1 \) as a Topological/Lie group on the n-sphere \( S^n \). The action is defined for odd-dimensional spheres, specifically through the mapping \( e^{i\phi} \cdot (z_1, \ldots, z_k) = (e^{i\phi}z_1, \ldots, e^{i\phi}z_k) \), which preserves antipodal points. This action serves as a counterexample to demonstrate that odd-dimensional real projective spaces \( \mathbb{RP}^n \) lack the fixed point property, meaning there exist continuous self-maps that do not fix any element. The conversation also touches on the implications of this action in terms of homotopy and fixed point properties.

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  • Understanding of Topological and Lie groups, specifically \( S^1 \) and \( S^n \)
  • Familiarity with the concept of fixed point properties in topology
  • Knowledge of complex numbers and their geometric interpretations
  • Basic understanding of projective spaces, particularly \( \mathbb{RP}^n \)
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  • Research the properties of the antipodal map on spheres and its implications in topology
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Mathematicians, particularly those specializing in topology, algebraic topology, and geometric group theory, as well as students studying the properties of spheres and projective spaces.

  • #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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