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

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