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

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