INTRODUCTION xi
(3) Products of homogeneous spheres
Ki/Hi x K2/H2 =
SUl
x
S"2
where K\ /H\ is one of the spaces
SO(n)JSO(n - 1) = S""1
SU(n)/SU(n-l) =S2n~1
Sp(n)/Sp(n - 1) = S4""1
G2/SU(3) =
S6
Spin(T)/G2 =
S7
Spin(9)/Spin(7) =
SL5
and K2/H2 is one of the spaces
SO(2n)/SO(2n-l)=S
2 r a
-
1
SU(n)/SU(n - 1) = S 2 "" 1
Sp(ra)/Sp(n-1) = S 4 u " 1
Spin(7)/G2 =
§7
Spin(9)/Spin(7) = S15
(4) Some sporadic spaces
E
6
/F
4
(9,17)
Spin(10)/Spin(7) (9,15)
Spin(9)/G2 =
V2(Q2)
(7,15)
Spin(8)/G2 =
§7
x
S7
(7, 7)
SU(6)/Sp(3) = SU(5)/Sp(2) (5,9)
Spin(10)/SU(5) = Spin(9)/SU(4) (6,15)
Spin(7)/SU(3) = V2(W8) (6,7)
Sp(3)/Sp(l)xSp(L) (4,11)
Sp(3)/Sp(l)xH/3A1(SP(l))
(4,11)
SU(5)/SU(3) x SU(2) (4,9).
The proof proceeds as follows. We show first that G/H has the same rational
homotopy groups as the product
S"1
x
S712.
This homotopy theoretic result follows
from a generalization of a theorem by Cartan and Serre. The rational homotopy
groups of a compact Lie group G can be determined explicitly; they depend only
on the Dynkin diagram of G and the rank of the central torus. In particular, a
compact connected Lie group has the same rational homotopy groups as a product
of odd-dimensional spheres. It follows in our situation that rk(G) rk(H) e {1,2},
depending on whether n\ is even or odd. Using this fact, we determine the rational
Leray-Serre spectral sequence of the principal bundle
H —» G G/H.
If n\ is odd, then the spectral sequence collapses, and G has the same rational
cohomology as H x
S™1
x
S™2.
If n\ is even, then there are non-zero differentials
in the spectral sequence and the situation is more complicated. In both cases we
obtain a relation between the numbers (ni,ri2) and the degrees of the primitive
elements in the rational cohomology of G and H.
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