FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 17

Proof Maps

f.. :

M

---+

M

are defined exactly as in the proofs of Proposi-

tions 5.4 and 5.5. The hypothesis Hom(V1,

W)

i-

0 allows that

[!..]

E

f#oo(M).

The hypothesis of formality ensures that

[!.]

E

£* (M). Finally, the condition

Hom -

1

(V0

,

H* ( M))

=

0 guarantees that distinct

..

yield distinct homotopy classes

[f.].

0

5.9 Remark Because of the equivalence of the homotopy category of minimal

algebras and the homotopy category of rational spaces, the results of this section for

the subgroups of £(M) have counterparts as results for the appropriate subgroups

of £(A), where A is a rational space (see Remark 2.3). Furthermore, by Remark

3.1, the graded vector spaces V0 and V1 which appear in the hypotheses in this

section can be identified with the image and kernel of the Hurewicz homomorphism

h. : 1r#(A)

---+

H.(A) of the rational space A. We leave to the reader the explicit

restatements of the results of this section. In addition, we can carry this analysis

further by considering finite, simple complexes X of dimension N. Let M be the

minimal model of X and suppose that £#N(M)

=

1.

Since £#N(M) and £#(XQ)

are anti-isomorphic (Remark 2.3), £#(XQ) is trivial. By Proposition 2.1, £#(X)

is finite. Likewise, if £#N(M) is infinite, then£# (X) is infinite. Similarly, if

X

is

1-connected, £*(M) is trivial or infinite according as £.(X) is finite or infinite. A

similar discussion holds for £*# (X).

Section 6 - Generalized Flag Manifolds

In this section we illustrate the obstruction theory of §§3 and 4 with several

examples in which we determine the finiteness of£. (X) and £#(X) for a generalized

flag manifold X.

Let U(n) denote the unitary group, then the homogeneous space

X= U(n)j(U(n

1 )

x · · · x

U(nk)) ,

where

n1

+ · · · +

nk

~

n and

n1 ~ · · · ~

nk, is called a generalized flag manifold.

This terminogy comes from the fact that, when n

1

+ · · · +

nk

=

n, X is a flag

manifold. We consider the space

X

in several examples below and so begin with a

brief discussion of its minimal model.

According to [G-H-V,p.476] the following (non-minimal) DG algebra gives

Proof Maps

f.. :

M

---+

M

are defined exactly as in the proofs of Proposi-

tions 5.4 and 5.5. The hypothesis Hom(V1,

W)

i-

0 allows that

[!..]

E

f#oo(M).

The hypothesis of formality ensures that

[!.]

E

£* (M). Finally, the condition

Hom -

1

(V0

,

H* ( M))

=

0 guarantees that distinct

..

yield distinct homotopy classes

[f.].

0

5.9 Remark Because of the equivalence of the homotopy category of minimal

algebras and the homotopy category of rational spaces, the results of this section for

the subgroups of £(M) have counterparts as results for the appropriate subgroups

of £(A), where A is a rational space (see Remark 2.3). Furthermore, by Remark

3.1, the graded vector spaces V0 and V1 which appear in the hypotheses in this

section can be identified with the image and kernel of the Hurewicz homomorphism

h. : 1r#(A)

---+

H.(A) of the rational space A. We leave to the reader the explicit

restatements of the results of this section. In addition, we can carry this analysis

further by considering finite, simple complexes X of dimension N. Let M be the

minimal model of X and suppose that £#N(M)

=

1.

Since £#N(M) and £#(XQ)

are anti-isomorphic (Remark 2.3), £#(XQ) is trivial. By Proposition 2.1, £#(X)

is finite. Likewise, if £#N(M) is infinite, then£# (X) is infinite. Similarly, if

X

is

1-connected, £*(M) is trivial or infinite according as £.(X) is finite or infinite. A

similar discussion holds for £*# (X).

Section 6 - Generalized Flag Manifolds

In this section we illustrate the obstruction theory of §§3 and 4 with several

examples in which we determine the finiteness of£. (X) and £#(X) for a generalized

flag manifold X.

Let U(n) denote the unitary group, then the homogeneous space

X= U(n)j(U(n

1 )

x · · · x

U(nk)) ,

where

n1

+ · · · +

nk

~

n and

n1 ~ · · · ~

nk, is called a generalized flag manifold.

This terminogy comes from the fact that, when n

1

+ · · · +

nk

=

n, X is a flag

manifold. We consider the space

X

in several examples below and so begin with a

brief discussion of its minimal model.

According to [G-H-V,p.476] the following (non-minimal) DG algebra gives