By Mazurov V. D.
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Extra resources for 2-Signalizers of finite groups
In the smooth case, the slice representations (Sx is a cell, normal to the orbit at x) are smoothly equivalent to linear representations, and it can easily be shown that as linear representations, they are linearly equivalent. If Gx = 1, then Qw = ψ(Gx ) ∼ = Gx /Gx and (Gx , Sx ) factors as Gx \ (H, Sx ) =(Gx , Sx ) −−−−→ (Gx /Gx , Gx \Sx )= (ψ(Gx ), Σw ) ψ(G )\ ν x G x (Gx , Sx ) −−−− → Gx \Sx It is significant to note that when (G, XK ) is a principal action, then the stabilizers of the Q-action on W encodes all the slice information of (G, X).
However, the extended action (S 1 , S 2n−1 ) has finite isotropy but is not locally injective. 13 Proposition. Suppose P is a principal G-bundle where G is a connected Lie group, and Π ⊂ TOPG (P ) is a group of covering transformations of P acting properly, that centralizes (G) and (G) ∩ Π = 1. Then the induced G-action on Π\P = X is locally injective. Proof. Since Π commutes with (G), there is induced a G-action on Π\P = X, which is covered by (G) on P . Because G is connected, this lift is the unique lift to P covering the induced G-action on X.
Since Gy is finite, Gy = 1G . 12 Proposition. If (G, X) is a locally injective G-action, then the lifted Gaction (G, XIm(evx∗ ) ) is a free action and conversely. Moreover, if G is a Lie group acting properly on X, then (G, XIm(evx∗ ) ) is proper so the action is a principal Gaction on XIm(evx∗ ) . Proof. Pick a base point x ∈ X and let H = Im(evx∗ ). Then (G, X) lifts to (G, XH ) and put x to be the path class of the constant loop at x. We have seen ψ(Gx ) = Qρ(x) ⊂ π1 (X, x)/H. The kernel is easily seen to be Gx but this by hypothesis is trivial.