FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 19
with
lvi,j I
=
2j and
lwi I
=
2i - 1. D
We now apply the results of §5 to these spaces.
6.2 Proposition Let X= U(n)j(U(nt)
x · · · x
U(nk)). Then £*(Mx)
=
1, and
hence £*(X) is finite.
Proof From the description of the minimal model in Lemma 6.1, we see that the
cohomology of X is evenly graded in degrees through the highest degree generator in
V1. Hence Hom(V1 ,H*(X;Q))
=
0. The proposition now follows from Proposition
5.1 and Remark 5.9 since X is 1-connected. D
Note that this extends Remark 5.2 to a non-maximal rank homogeneous space.
Next we discuss£# of these spaces.
6.3 Proposition If X= U(n)j(U(n1)
x ..
·xU(nk)) with r
=
n-(n1
+ ..
+nk)
2::
2,
then £#(X) is infinite.
Proof We show is infinite, forM the minimal model of X, by constructing
an explicit automorphism of M. Using the notation of 6.1, we define
f.. :
M
---t
M
by
where
.
is a nonzero rational and
f..
=
~
on all other generators. Clearly
[/..]
E
In cohomology,
Since
.
=I
p,
implies
J;
=I 1;,
the elements
[f..]
are distinct for different
..
Thus
(M) is infinite. D
We now establish some results in which £# is finite. For this we restrict to the
case
k
=
2.
6.4 Proposition If n
=
n1
+
n2, then
(i) £#(U(n)j(U(nt) x U(n2))) is finite and
(ii) £#(U(n
+
1)/(U(nt) x U(n2))) is finite.
Proof (i) Let X= U(n)j(U(nt) x U(n2)) and let M be the minimal model of X.
We show that £#N(M)
=
1, where N is the dimension of X. It is sufficient to
show by Proposition 6.2 that if[/]
E
£#N(M), then
f*
=" :
H*(M)
---t
H*(M).
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