Compact Connected Lie Transformation Groups on Spheres With by Eldar Straume

By Eldar Straume

During this ebook, the writer incorporates out a scientific research and development of all attainable differentiable (homotopy) G-spheres with 2-dimensional orbit house, the place G is any compact hooked up Lie crew. according to the geometric weight method category of half I, the potential orbit buildings are made up our minds, and the unique G-spheres are built by means of equivariant twisting of the orthogonal versions.

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The smoothness of the homeomorphisms involved will depend crucially on the next lemma. Let (H c L) be a spherical pair. It is well known that there is a unique representation *F : L —> 0(m+l) of cohomogeneity cQ¥) = 1, and with H as a principal isotropy group. Via this action of L we shall identify L/H with the unit sphere S m . 1 Let S m = L/H be as above and let G = L in (5). Then the corresponding homomorphism p in (5) maps each automorphism to an isometry, namely p : Aut(H; L) -> 0(m+l) c Diff(S m).

By [S4, p. 12], hence the last case is actually impossible. LOW COHOMOGENEITY ACTIONS 29 Inn(-; K) c Aut(-; K) denotes the subgroup of inner automorphisms, and Out(-; K) is the quotient group. 10 p maps Inn(H, L Q , L j ; K) into 0(q+l) if and only if R maps N K ( H , L 0 , L]) into 0(q+l). More precisely, let p n be the image of n by the composition N K ( H, L 0 , L i ) -> Inn(H, L 0 , L j ; K) P -> Diff^S^), where the first map is the natural surjective homomorphism. Then p n = 0(q+l ) is the given representation, cf.

2), (6). , and assume also Aut(H, L 0 , L j ; K) e Aut(K'; K). 8 K is p -determined by its connected component K°. Proof By the above observation we shall assume K and K° are not C-equivalent. Then K/K° acts by reflection on the arc I' = S^/K0 and one of the two halves may be identified with I. 7 applies to this pair. Moreover, K' is a group of index 2 in K, say K = < K', y > where y 2 e K'. It remains to show that K is p -determined by K'. In terms of cross sections and associated triples we can make the following choices, namely H = H', L 0 = L' 0 , L j = < H, y > , L'j = yLQy" .

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