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.
Previous Page Next Page