14
MARTIN ARKOWITZ AND GREGORY LUPTON
5.2 Remark This proposition applies to all of the spaces listed in Remarks 3.7.
In particular, if
A
is the rationalization of a homogeneous space
G
I
H
with
H
~
G
of maximal rank, then f* (MA)
=
1, and so t'* (A)
=
1.
Thus by Proposition
2.1, t'*(GIH) is finite if GIH is 1-connected. If GIH is not 1-connected then the
conclusion follows since the function [G
I
H, G
I
H] ---+ [A, A] is finite-to-one [H-M-
R,Cor.5.4]. This generalizes a proposition of Notbohm-Smith [N-S,Prop.3.1]
who proved this result in the case H is a maximal torus. For some non-maximal
rank cases where
t'* ( G I H) is finite, see §6.
We next consider when f* (M) is infinite. Here we add the hypothesis of
formality.
5.3 Lemma If M
=
A(Vo,
V1
;
d) is formal and
f :
M
-l
M is a map such that
!Iva= tlv0
,
then
f*
=
L:
H*(M)----+ H*(M).
Proof Since
}vt
is 2-stage,
M
is a bigraded DG algebra, with second grading given
by word length in the generators of vl' i.e.'
M
=
l:i~O
Mi'
where
The differential
d
decreases second grading by one. The second grading passes to
the cohomology H*(M), making it a bigraded algebra. Then
f
induces
f*
which
is the identity on lower degree 0, i.e.,
f*IH5(M)
=
L:
H0(.A1.)---+ H0(M).
Since
M
is formal and
d :
V1
----+
A(Vo)
is injective, we have H* (M)
=
H(j (M) [D-G-M-
S,Thm.4.1]. Thus
f* =
t:
H*(M)----+ H*(M). D
We now give conditions for t'* (M) to be infinite.
5.4 Proposition Let
M =
A(V0
,
V1 ;d) be formal. IfHom-
1
(V0 ,H*(M))
=
0 and
Hom(V1
,
H*(M))
f.
0, then t'*(M) is infinite.
Proof We construct infinitely many elements off* ( M).
By
hypothesis, there exists
a positive integer
l
such that
V{
f.
0 and
H
1
(M)
f.
0. Now choose a basis element
wk
E V1 and cocycle x E
M
which is not a coboundary, both of the same degree
l.
If
A
is a nonzero rational,
A
E Q*, define a map
J;., : M
---+
M
as follows. Set
and
f..
=
Lon
all other basis elements of
V.
Furthermore, since
M
is formal,
f~
=
L
by Lemma 5.3. Finally, if.. and
11
are distinct nonzero rationals, then
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