2 MARTIN ARKOWITZ AND GREGORY LUPTON
finiteness results for £*(X) and for £#(X) in [Ma1] and [Ma3]. In [Au-Le] Aubry
and Lemaire study the set of equivalences which induce the identity on all homology
and homotopy groups. From their work it follows that £*#(83 x 8
3
)
has at least
144 elements and that £*#((85 x 8
5
)
V 8
5
V 8
6
)
is infinite.
We next describe the organization of the paper and our results. After a pre-
liminary section on notation, we qiscuss in §3 and §4 an obstruction theory for ho-
motopy of homomorphisms of minimal differential graded algebras. We restrict our
attention to a certain class of minimal differential graded algebras, the 2-stage alge-
bras. A minimal algebra M with differential dis called 2-stage if M
=
A(V
0
EB VI),
the free commutative algebra on the direct sum of graded vector spaces
Vo
and V1,
such that
diva
=
0 and dlv1
:
V1
--t
A(Vo) (see §3). This class of algebras includes
the minimal models of homogeneous spaces. Our obstruction theory deals with
homomorphisms
J,
g :
M
--t
N,
where
M
is 2-stage and
N
is an arbitrary minimal
algebra. If
!Iva= glva,
an obstruction 0 1(f,g):
V1
--t
H*(N) is defined. We show
in §3 that
f
is homotopic to
g
if 01
(!,g)
=
0. In §4 we give conditions under
which the converse of this result holds. This obstruction theory is of interest in its
own right and can be used to obtain results about the set of homotopy classes of
maps [X, Y] when Y has a 2-stage minimal model. We illustrate this use of the
theory in Remarks 3. 7 and Corollary 4.9. In §5 we focus on the subgroups of the
group £(M) of homotopy classes of homotopy equivalences of a minimal algebra
M which are analogous to the subgroups of £(X) mentioned earlier. We use our
obstruction theory to give fairly general conditions on M in terms of V0
,
V1
and
H*(M) under which these subgroups of £(M) are finite or infinite. This enables
us to obtain corresponding results for the subgroups of £(XIQI), where XIQI is the
rationalization of X and M is the minimal model of X. By the technique of dera-
tionalization we then obtain finiteness results for the subgroups £*(X), £#(X) and
£*#(X) of £(X). This method is illustrated in the last two sections of the paper
where we make a detailed analysis of the subgroups of £(M) forM the minimal
model of certain classes of spaces
X.
In §6 we take
X
to be a generalized flag
manifold U(n)/(U(nl) x · · · x U(nk)) and investigate £*(X) and £#(X). We show
that £*(X) is always finite and have some results on the finiteness of £#(X). In §7
we take
X
to be a product of spheres and refine the obstruction theory of §§3 and
4 in this case. We then obtain necessary and sufficient conditions for £*#(X) to be
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