14

MARTIN ARKOWITZ AND GREGORY LUPTON

5.2 Remark This proposition applies to all of the spaces listed in Remarks 3.7.

In particular, if

A

is the rationalization of a homogeneous space

G

I

H

with

H

~

G

of maximal rank, then f* (MA)

=

1, and so t'* (A)

=

1.

Thus by Proposition

2.1, t'*(GIH) is finite if GIH is 1-connected. If GIH is not 1-connected then the

conclusion follows since the function [G

I

H, G

I

H] ---+ [A, A] is finite-to-one [H-M-

R,Cor.5.4]. This generalizes a proposition of Notbohm-Smith [N-S,Prop.3.1]

who proved this result in the case H is a maximal torus. For some non-maximal

rank cases where

t'* ( G I H) is finite, see §6.

We next consider when f* (M) is infinite. Here we add the hypothesis of

formality.

5.3 Lemma If M

=

A(Vo,

V1

;

d) is formal and

f :

M

-l

M is a map such that

!Iva= tlv0

,

then

f*

=

L:

H*(M)----+ H*(M).

Proof Since

}vt

is 2-stage,

M

is a bigraded DG algebra, with second grading given

by word length in the generators of vl' i.e.'

M

=

l:i~O

Mi'

where

The differential

d

decreases second grading by one. The second grading passes to

the cohomology H*(M), making it a bigraded algebra. Then

f

induces

f*

which

is the identity on lower degree 0, i.e.,

f*IH5(M)

=

L:

H0(.A1.)---+ H0(M).

Since

M

is formal and

d :

V1

----+

A(Vo)

is injective, we have H* (M)

=

H(j (M) [D-G-M-

S,Thm.4.1]. Thus

f* =

t:

H*(M)----+ H*(M). D

We now give conditions for t'* (M) to be infinite.

5.4 Proposition Let

M =

A(V0

,

V1 ;d) be formal. IfHom-

1

(V0 ,H*(M))

=

0 and

Hom(V1

,

H*(M))

f.

0, then t'*(M) is infinite.

Proof We construct infinitely many elements off* ( M).

By

hypothesis, there exists

a positive integer

l

such that

V{

f.

0 and

H

1

(M)

f.

0. Now choose a basis element

wk

E V1 and cocycle x E

M

which is not a coboundary, both of the same degree

l.

If

A

is a nonzero rational,

A

E Q*, define a map

J;., : M

---+

M

as follows. Set

and

f..

=

Lon

all other basis elements of

V.

Furthermore, since

M

is formal,

f~

=

L

by Lemma 5.3. Finally, if.. and

11

are distinct nonzero rationals, then

MARTIN ARKOWITZ AND GREGORY LUPTON

5.2 Remark This proposition applies to all of the spaces listed in Remarks 3.7.

In particular, if

A

is the rationalization of a homogeneous space

G

I

H

with

H

~

G

of maximal rank, then f* (MA)

=

1, and so t'* (A)

=

1.

Thus by Proposition

2.1, t'*(GIH) is finite if GIH is 1-connected. If GIH is not 1-connected then the

conclusion follows since the function [G

I

H, G

I

H] ---+ [A, A] is finite-to-one [H-M-

R,Cor.5.4]. This generalizes a proposition of Notbohm-Smith [N-S,Prop.3.1]

who proved this result in the case H is a maximal torus. For some non-maximal

rank cases where

t'* ( G I H) is finite, see §6.

We next consider when f* (M) is infinite. Here we add the hypothesis of

formality.

5.3 Lemma If M

=

A(Vo,

V1

;

d) is formal and

f :

M

-l

M is a map such that

!Iva= tlv0

,

then

f*

=

L:

H*(M)----+ H*(M).

Proof Since

}vt

is 2-stage,

M

is a bigraded DG algebra, with second grading given

by word length in the generators of vl' i.e.'

M

=

l:i~O

Mi'

where

The differential

d

decreases second grading by one. The second grading passes to

the cohomology H*(M), making it a bigraded algebra. Then

f

induces

f*

which

is the identity on lower degree 0, i.e.,

f*IH5(M)

=

L:

H0(.A1.)---+ H0(M).

Since

M

is formal and

d :

V1

----+

A(Vo)

is injective, we have H* (M)

=

H(j (M) [D-G-M-

S,Thm.4.1]. Thus

f* =

t:

H*(M)----+ H*(M). D

We now give conditions for t'* (M) to be infinite.

5.4 Proposition Let

M =

A(V0

,

V1 ;d) be formal. IfHom-

1

(V0 ,H*(M))

=

0 and

Hom(V1

,

H*(M))

f.

0, then t'*(M) is infinite.

Proof We construct infinitely many elements off* ( M).

By

hypothesis, there exists

a positive integer

l

such that

V{

f.

0 and

H

1

(M)

f.

0. Now choose a basis element

wk

E V1 and cocycle x E

M

which is not a coboundary, both of the same degree

l.

If

A

is a nonzero rational,

A

E Q*, define a map

J;., : M

---+

M

as follows. Set

and

f..

=

Lon

all other basis elements of

V.

Furthermore, since

M

is formal,

f~

=

L

by Lemma 5.3. Finally, if.. and

11

are distinct nonzero rationals, then