MARTIN ARKOWITZ AND GREGORY LUPTON
5.2 Remark This proposition applies to all of the spaces listed in Remarks 3.7.
In particular, if
is the rationalization of a homogeneous space
of maximal rank, then f* (MA)
1, and so t'* (A)
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
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
5.3 Lemma If M
d) is formal and
M is a map such that
is a bigraded DG algebra, with second grading given
by word length in the generators of vl' i.e.'
decreases second grading by one. The second grading passes to
the cohomology H*(M), making it a bigraded algebra. Then
is the identity on lower degree 0, i.e.,
is formal and
is injective, we have H* (M)
H(j (M) [D-G-M-
H*(M)----+ H*(M). D
We now give conditions for t'* (M) to be infinite.
5.4 Proposition Let
V1 ;d) be formal. IfHom-
0, then t'*(M) is infinite.
Proof We construct infinitely many elements off* ( M).
hypothesis, there exists
a positive integer
0. Now choose a basis element
E V1 and cocycle x E
which is not a coboundary, both of the same degree
is a nonzero rational,
E Q*, define a map
J;., : M
as follows. Set
all other basis elements of
by Lemma 5.3. Finally, if.. and
are distinct nonzero rationals, then