FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 9
3.6 Corollary If
M
= A(V0
,
V1;
d)
is a 2-stage, formal minimal algebra,
N
is any
formal minimal algebra and Hom(V1, H*
(N))
= 0, then
I: [M,N] ____.
Hom(H*(M), H*(N))
is a bijection.
Proof
It
is well known (see, e.g., [Ar]) that
I
is onto when
M
and
N
are formal.
The result now follows from Corollary 3.5 D
3. 7 Remarks These results immediately translate into results for spaces. If A
and
B
are nilpotent rational spaces of finite type and if the minimal model
ME
is
2-stage with Hom(V1, H* (A;
Q))
= 0 then
I: [A,B]___.
Hom(H*(B;Q),H*(A;Q))
is one-one by Corollary 3.5. In particular, if
ME
is 2-stage with V1 oddly graded and
H* (A;
Q)
evenly graded, then the hypothesis Hom(V1, H* (A;
Q))
= 0 holds, and so
I is one-one. For example, if B is the rationalization of SU(6)I(SU(3) x SU(3)),
then
ME
is 2-stage with
Vo
=
(y4,Y6l
and V1 =
(x7,xg,xn),
with subscripts
denoting degrees (see §6). If A is such that Hi(A;
Q)
= 0 fori= 7, 9, 11, then
I
is
one-one.
If
A
and
B
are rational spaces as above, and in addition both are formal
with, for example, 2-stage minimal model having V1 oddly graded and cohomology
evenly graded, then
I
is a bijection by Corollary 3.6. Examples of such spaces are
the rationalizations of the following spaces :
(i) Homogeneous spaces GIH, where G is a connected Lie group and H
~
G is a
closed, connected subgroup of maximal rank (see [H-T]).
(ii) Even dimensional spheres S
2n.
(iii) Products of spaces in (i) and (ii).
Thus, for example, if GIH and G'IH' are homogeneous spaces as in (i), then
I: [(GIH)Q,(G'IH')Q]___. Hom(H*(G'IH';Q),H*(GIH;Q))
is a bijection. This result has been proved by Glover and Homer [G-H,Thm.l.l]
in the case G
I
H and G'
I
H' are complex or quaternionic flag manifolds.
These considerations can be extended from rational spaces to spaces in the
following way. If
X
and
Y
are nilpotent spaces of finite type such that
X
has the
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