Timeline for How to extend an equivariant map from a compact Lie group
Current License: CC BY-SA 3.0
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| Oct 7, 2015 at 9:46 | comment | added | Sebastian Goette | If you follow the approach I outlined, you will need some technology. So you could follow Strickland's suggestion and consider P- or PxH-CW structures on X and G. Those also start with small cells with large stabilisers. I don't know which references are good, but maybe you start with Matumoto, Takao, On G-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363–374, or Illman, Sören Equivariant singular homology and cohomology for actions of compact Lie groups, Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, 403–415. | |
| Oct 7, 2015 at 1:24 | history | edited | Megan |
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| Oct 7, 2015 at 1:16 | comment | added | Megan | Thanks. But I'm not familiar with what you said. Why the points with the greatest stabilizers give the boundary values for the subspaces with smaller ones? So if $C_G(g)\backslash G/H$ does not have generic point, we just end till reach the points with the smallest stabilizer group? BTW, does $C_G(g)\backslash G/H$ usually have generic points.... I appreciate a lot if you can recommend some reference. | |
| Oct 6, 2015 at 19:28 | answer | added | Neil Strickland | timeline score: 2 | |
| Oct 6, 2015 at 18:03 | comment | added | Sebastian Goette | ... The points in C\G/H come with stabilizers aHa^{-1}\cap C. These groups are ordered partially by inclusion. I would suggest to fix f for those points with greatest stabilizers first. Then you get boundary values for the subspaces with smaller stabilisers, and you can work your way up to the generic points with the smallest possible stabilisers. This should produce a continuous map f for you. | |
| Oct 6, 2015 at 17:56 | comment | added | Sebastian Goette | From your formulas, it seems that the centralizer acts from the left on G, and H acts only on G, and from the right. Is this correct? Next, to be more precise, you say that a map f from your space is determined by its value on one representative of each double coset, and you want this value to lie in the fixpoint set you specified. | |
| Oct 6, 2015 at 17:18 | history | edited | Megan | CC BY-SA 3.0 |
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| Oct 6, 2015 at 17:12 | history | asked | Megan | CC BY-SA 3.0 |