LOW COHOMOGENEITY ACTIONS 11
( 8 ) c(M-a ®0 ^b ®0 M-c) 7, V a, b, c 2.
We are left with the subcases a) q = 2, and either m = 1, or m = 2 and F = IR, or
b) q 3 and G has a central circle group.
Consider first the situation where O = §i®f O', G = K(1) x G1, G' is semisimple and
F = 0 or IH. Observe that
0, 2, 3 if F = I H
Due to the simple observation
c(o') = 1 ^ c(818)|p O') = 1 (cf. Table I ),
we also conclude that
c(o') = 2 = c f s ^ f p *') = 2.
Assume G' is simple. In Case (i) we found those linear groups (G',^) with
c(o') 4 or 6 for £' complex or quatemionic, respectively. Now it is easy to find
those cases where c(O) 3. On the other hand, if G' has s 2 simple factors then F = G,
and by (8) we conclude s = 2. In fact, the only possibilities are O = [i2 ®0 M-k
\i2 ®0 v
k
and
II
3
®
0
MR-
Finally, let G = SO(2)xG\ O = p2 ^ O ' . From the relations
c(p
2
® R O ' ) = c(20') - 1 2c(0') - 1,
and the previous knowledge of those (G',01) with £' irreducible and c(O') = 2, we find
that c(O') = 1 is a necessary condition. The possible candidates
(G',01)
satisfying
c(20') 4 are easily checked.
Case (iii) G nonsimple, O = Oj + . . . + O^, k = 2 or 3. Here the diagram r(O) has
k simplices and c(0) 3 implies X c(Oj) 3.
Assume first G is semisimple. The problem is to combine two or three d-simplices,
d 0, into a connected diagram r(£) of cohomogeneity 3, and the only difficulty is to
estimate dim G^. Here (6) may be useful. One finds that each simplex is at most 1-
dimensional, and r(O) - {0-simplices} is contractible. Furthermore, all vertices are
of type Sp(m), m 1, except possibly one vertex of type SU(m), m 2, cf. (4 ).
( 9 ) c^c&jp O') = c(E') - e,
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