20

MARTIN ARKOWITZ AND GREGORY LUPTON

We modify the notation in 6.1 and write M =A( vi, ... ,

Vn

1

,

Wn

2

+I, ... ,

Wn)

with

I

vi

I

= 2i and

lwj

I

= 2j-

1.

Ifni = 1, the argument is trivial. Ifni

~

2, then there

are at least two generators VI and v2 of

M.

Thus

f(vl)

=VI and

j(v2)

= v2 +

.vr

for some).. E

IQ.

Suppose ni n2. On cohomology, f*[vi] =[vi]· Hence by a result

of Hoffman [Ho,Thm.l.l],

f*

is the identity. Now suppose ni = n2. Then, by the

same result of Hoffman, either

f*

is the identity or f*[v2]

==

-[v2] + [viF· But the

latter equation contradicts the fact that [/] E E#N(M). Thus

f*

=

i.

By Remark

5.9, E#(X) is finite. This completes the proof of (i).

(ii)

Now let X= U(n+ 1)/(U(ni) x U(n

2

)),

where N is the dimension of X, and let

Mx be the minimal model of X. Then Mx

~

M0A(w), where M is as in (i) and

lwl = 2n+

1.

Since E*(Mx) = 1 by 6.2, it suffices to show that if[/]

E

E#N(Mx),

then

f*

is the identity. But g

=JiM :

M

---

M since the generators of M are in

degrees 2n +

1.

By (i), g* =

i.

Now f(w) = w +X for

[x]

E H 2n+I(M). But

H*(M) is evenly graded and so f*[w] = [w]. It follows that

f*

=

i.

Therefore

E#(X) is finite by Remark 5.9. D

6.5 Remark Note that U(n)/(U(ni) x U(n

2

))

is just the complex Grassmanian

G( ni, ni + n2) of ni-dimensional planes in complex n-dimensional space,

n

= ni +

n2. In Proposition 6.4 we have restricted our attention to the case k = 2. This

was to allow us to use Hoffman's result. It has been conjectured that analogues of

Hoffman's result hold for other flag manifolds- see the remark in [G-H,p.425].

When a suitable result holds with

k

?:

3, then an argument similar to the argument

of Proposition 6.4 would show that E#(X) is finite.

In the above examples we have used information about the homogeneous spaces

U(n)/(U(ni) x · · · x U(nk)) and their minimal models to obtain information on the

group of equivalences. Similar information about other homogeneous spaces is

available

[G~H-

V] and can be used to obtain similar results where appropriate.

Section 7 - Products of Spheres

In this section we use the methods of §§3 and 4 to determine when the group

£*#(X) is finite and when it is infinite in the case that X is a product of spheres. We

begin by fixing our notation. Let

X

=

sml

X ... X

smp

X

snl

X ••• X

snq' where

ffij

is

odd and nk is even. If M is the minimal model of X, then Proposition 5.6 applies to

M and we have that E#n(M) =

£#

00

(M), for any n

~

mi +· · ·+mp+ni +· · ·+nq.

MARTIN ARKOWITZ AND GREGORY LUPTON

We modify the notation in 6.1 and write M =A( vi, ... ,

Vn

1

,

Wn

2

+I, ... ,

Wn)

with

I

vi

I

= 2i and

lwj

I

= 2j-

1.

Ifni = 1, the argument is trivial. Ifni

~

2, then there

are at least two generators VI and v2 of

M.

Thus

f(vl)

=VI and

j(v2)

= v2 +

.vr

for some).. E

IQ.

Suppose ni n2. On cohomology, f*[vi] =[vi]· Hence by a result

of Hoffman [Ho,Thm.l.l],

f*

is the identity. Now suppose ni = n2. Then, by the

same result of Hoffman, either

f*

is the identity or f*[v2]

==

-[v2] + [viF· But the

latter equation contradicts the fact that [/] E E#N(M). Thus

f*

=

i.

By Remark

5.9, E#(X) is finite. This completes the proof of (i).

(ii)

Now let X= U(n+ 1)/(U(ni) x U(n

2

)),

where N is the dimension of X, and let

Mx be the minimal model of X. Then Mx

~

M0A(w), where M is as in (i) and

lwl = 2n+

1.

Since E*(Mx) = 1 by 6.2, it suffices to show that if[/]

E

E#N(Mx),

then

f*

is the identity. But g

=JiM :

M

---

M since the generators of M are in

degrees 2n +

1.

By (i), g* =

i.

Now f(w) = w +X for

[x]

E H 2n+I(M). But

H*(M) is evenly graded and so f*[w] = [w]. It follows that

f*

=

i.

Therefore

E#(X) is finite by Remark 5.9. D

6.5 Remark Note that U(n)/(U(ni) x U(n

2

))

is just the complex Grassmanian

G( ni, ni + n2) of ni-dimensional planes in complex n-dimensional space,

n

= ni +

n2. In Proposition 6.4 we have restricted our attention to the case k = 2. This

was to allow us to use Hoffman's result. It has been conjectured that analogues of

Hoffman's result hold for other flag manifolds- see the remark in [G-H,p.425].

When a suitable result holds with

k

?:

3, then an argument similar to the argument

of Proposition 6.4 would show that E#(X) is finite.

In the above examples we have used information about the homogeneous spaces

U(n)/(U(ni) x · · · x U(nk)) and their minimal models to obtain information on the

group of equivalences. Similar information about other homogeneous spaces is

available

[G~H-

V] and can be used to obtain similar results where appropriate.

Section 7 - Products of Spheres

In this section we use the methods of §§3 and 4 to determine when the group

£*#(X) is finite and when it is infinite in the case that X is a product of spheres. We

begin by fixing our notation. Let

X

=

sml

X ... X

smp

X

snl

X ••• X

snq' where

ffij

is

odd and nk is even. If M is the minimal model of X, then Proposition 5.6 applies to

M and we have that E#n(M) =

£#

00

(M), for any n

~

mi +· · ·+mp+ni +· · ·+nq.