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
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