LOW COHOMOGENEITY ACTIONS 3
In fact, they actually serve as the guiding beacons which direct the whole course of the
classification task of this paper. Besides, it is useful to have such complete tables for
various other reasons, such as applications in equivariant differential geometry and
invariant theory of linear groups, cf. e.g., [S6], [NS].
Technically, this paper relies heavily on the three kinds of geometric weight
systems associated to a given action on an integral homology sphere (cf. Ch.ll, §1 ).They
provide the crucial basic invariants which enable one to reduce the complete
determination of orbit structures of low cohomogeneity transformation groups to that of
the explicit groups together with their weight systems. This is exactly the major task of
Chapter II; it constitutes the decisive step toward the classification theorems.
Conceptually, the definition of the integral (resp. rational or 2
p
-) weight system is
based upon the splitting theorem for the equivariant Euler class (cf. [Hs1]), which
proves the linear resemblance of the topological actions of tori (or p-tori) at the
homological level. Therefore, the whole approach of Chapter II can be considered as a
kind of "homological linearization" of differentiable transformation groups on spheres
via the maximal torus theorem and the splitting theorems.
We state the main results of this paper as the following three theorems.
Theorem A Let G be a compact connected Lie group and consider differentiable
actions of G on homology n -spheres X
n
~z S
n
with dim
Xn/G
3. Assume X
n
is not the
3-dimensional Poincare sphere with the transitive SO(3)-action. Then the integral
geometric weight system
Q(Xn)
is of linear type, that is, for each X
n
there is a
representation O : G - SO(n+1) such that
Q(Xn)
= Q(O), with the following exception
where dim Xn/G = 1:
G = SO(2) x SO(m), m = 2k+1 3,
H(X
n
) = {±6 } + {±(h9 ± ij)}, odd h 1, n = 4k+1 5 ,
and if n = 13, the restriction of
Q(Xn)
to the subgroup SO(2) x G
2
of SO(2) x SO(7).
(Here 0 is a unit weight of SO(2) and {±i:} are the nonzero weights of the standard
representation p
m
of SO(m) on (Rm.)
Theorem B Let X
n
~
2
S
n
be a homology sphere with a G-action whose weight
pattern
Q'(Xn)
coincides with the weight pattern of some linear representation of
cohomogeneity 3. Then the fixed point set of G is a Z^-homology sphere.
The weight pattern is the weight system minus its zero weights. We refer to
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