Theorem B-j in Chapter III for a more precise but technical formulation of the above
theorem. (In fact, a stronger version will be proved in [S5]). The above theorem is a
natural generalization of Montgomery-Yang [MY; ll,Theorem B]), where X
= S
the standard sphere, G has a fixed point and dim
= 3. They conclude that the fixed
point set is a sphere of dimension 2.
Theorem C Let X
be a homology sphere which is a compact smooth G-
manifold of cohomogeneity one, that is, dim Xn/G = 1. Then either X n is equivariantly
diffeomorphic to the standard sphere S
with some orthogonal G-action, or X
is a homotopy sphere Z and (G, Q(E )) is one of the exceptional cases in Theorem A.
(See Remark below.)
Remark The pairs (G,Z ) are exactly the missing cases in [W]. As is well known,
each (G, X ) can be presented as the following G-invariant, codimension 2 subvariety
of the unit sphere S
n + 2
in C
2 k + 2
= R
4 k + 4
, n = 4k+1 5 :
) h + (
Z 1
) 2 + ( z
2 k + 1
) 2 = 0, 2 IZjl2 = 1, h 1 is odd,
where G = SO(2)xSO(2k+1) (or SO(2)xG2, if k = 3 ) acts orthogonally on S
n + 2
the representation (jij) 2 + (ia1)h ®^( p2k+l)(E o n ^ 2 k + 2 - Note that the G-manifold E
is characterized by the odd integer h 1. (The case h =1 gives S
with orthogonal G-
action). We refer to [Br1], [Br2], [Bri], [Hi] for further information concerning the
construction of these G-manifolds.
We shall give some further comments and a brief description of our approach. First
of all, the exhaustive procedure followed in Chapter I also gives complete tables of
linear groups with higher cohomogeneities. One starts, of course, with simple linear
groups and these can be easily tabulated up to, say, cohomogeneity c 8, and for
semisimple groups the tables are still reasonably short in this range of c. However,
checking all non-semisimple groups as well is rather tedious even for c = 4 or 5.
The study of compact connected transformation groups on spheres is, also from our
point of view, considerably more difficult than in the euclidean case. The main reason is
the difference in the fixed point theory. Whereas a fixed point always exists for
euclidean G-spaces of low cohomogeneity, cf. [HS1], fixed points may not even exist for
the spherical linear models. In reality, this single fact essentially "explains" all the
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