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).

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).