3.6 Corollary If
is a 2-stage, formal minimal algebra,
formal minimal algebra and Hom(V1, H*
= 0, then
I: [M,N] ____.
is a bijection.
is well known (see, e.g., [Ar]) that
is onto when
The result now follows from Corollary 3.5 D
3. 7 Remarks These results immediately translate into results for spaces. If A
are nilpotent rational spaces of finite type and if the minimal model
2-stage with Hom(V1, H* (A;
= 0 then
is one-one by Corollary 3.5. In particular, if
is 2-stage with V1 oddly graded and
evenly graded, then the hypothesis Hom(V1, H* (A;
= 0 holds, and so
I is one-one. For example, if B is the rationalization of SU(6)I(SU(3) x SU(3)),
is 2-stage with
and V1 =
denoting degrees (see §6). If A is such that Hi(A;
= 0 fori= 7, 9, 11, then
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
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
(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
H and G'
H' are complex or quaternionic flag manifolds.
These considerations can be extended from rational spaces to spaces in the
following way. If
are nilpotent spaces of finite type such that