FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 13

show that if i

-::J

j, then

9i

and g1 are not homotopic. For this consider

by Lemma 4.2.

Thus 0

1

(gj,

gi)

#

0,

and so by Proposition 4. 7,

9i

and gj are not homotopic. D

To illustrate how these ideas apply to spaces, we prove the following result:

4.9 Corollary Let X be a 1-connected formal space of finite type and let Y be a

1-connected space of finite type with 2-stage minimal model A(Vo, V1;d). With the

above notation, if Hom-

1(V0

,H*(X;Q))

=

0 and Hom(V1 ,H*(X;Q))

#

0, then

the set [X, Y] is infinite.

Proof Let Mx and My be the minimal models of X andY. Then by Proposition

4.1 there exists a map

¢ :

My

-t

Mx, with corresponding map

g :

XIQI

-t

YIQI,

such that 0

1

( ¢, 0)

#

0 : V1

-t

H* (Mx). For any

t

E

Z, let

ft :

X

-t

X denote

a map such that ft(x)

=

tlxlx for all x

E

H*(X). By [Pa2,3.2], there is a map

h: X

-t

Y, and some map ft :X

-t

X, with

t

#

0, such that hi!J

=

gft

I!J·

From

this it follows that

0 1 (~,0):

V1

-t

H*(X;Q) is nonzero,

where~:

My

-t

Mx

is the map corresponding to hiQI. By [Sh] there is an integer s

#

0, ±1 such that

a map fs : X

-t

X exists. Let Bs : My

-t

Mx denote the map corresponding

to fs

I!J·

Then the obstructions 0 1

(0/

~'

0)

are all different for

r

=

1, 2, .... Thus

the maps

hf/

for

r

=

1, 2, ... represent distinct homotopy classes in [X,

Y],

by

Proposition 4.7. D

It is clear that there are many spaces X and Y which satisfy the hypotheses

of Corollary 4.9.

Section 5 - Subgroups of the Group of Homotopy Equivalences

In this section we use the obstruction theory of the previous two sections to

study subgroups of the group of homotopy equivalences. Our goal in this section

is to obtain fairly general results which hold for a large class of minimal algebras.

We begin with the subgroup £*(M) of £(M). Our first result is an immediate

consequence of Corollary 3.5:

5.1 Proposition Let M

=

A(V0

,

V1

;

d) be 2-stage with Hom(V1

,

H*(M))

=

0.

Then £*(M)

=

1. D

show that if i

-::J

j, then

9i

and g1 are not homotopic. For this consider

by Lemma 4.2.

Thus 0

1

(gj,

gi)

#

0,

and so by Proposition 4. 7,

9i

and gj are not homotopic. D

To illustrate how these ideas apply to spaces, we prove the following result:

4.9 Corollary Let X be a 1-connected formal space of finite type and let Y be a

1-connected space of finite type with 2-stage minimal model A(Vo, V1;d). With the

above notation, if Hom-

1(V0

,H*(X;Q))

=

0 and Hom(V1 ,H*(X;Q))

#

0, then

the set [X, Y] is infinite.

Proof Let Mx and My be the minimal models of X andY. Then by Proposition

4.1 there exists a map

¢ :

My

-t

Mx, with corresponding map

g :

XIQI

-t

YIQI,

such that 0

1

( ¢, 0)

#

0 : V1

-t

H* (Mx). For any

t

E

Z, let

ft :

X

-t

X denote

a map such that ft(x)

=

tlxlx for all x

E

H*(X). By [Pa2,3.2], there is a map

h: X

-t

Y, and some map ft :X

-t

X, with

t

#

0, such that hi!J

=

gft

I!J·

From

this it follows that

0 1 (~,0):

V1

-t

H*(X;Q) is nonzero,

where~:

My

-t

Mx

is the map corresponding to hiQI. By [Sh] there is an integer s

#

0, ±1 such that

a map fs : X

-t

X exists. Let Bs : My

-t

Mx denote the map corresponding

to fs

I!J·

Then the obstructions 0 1

(0/

~'

0)

are all different for

r

=

1, 2, .... Thus

the maps

hf/

for

r

=

1, 2, ... represent distinct homotopy classes in [X,

Y],

by

Proposition 4.7. D

It is clear that there are many spaces X and Y which satisfy the hypotheses

of Corollary 4.9.

Section 5 - Subgroups of the Group of Homotopy Equivalences

In this section we use the obstruction theory of the previous two sections to

study subgroups of the group of homotopy equivalences. Our goal in this section

is to obtain fairly general results which hold for a large class of minimal algebras.

We begin with the subgroup £*(M) of £(M). Our first result is an immediate

consequence of Corollary 3.5:

5.1 Proposition Let M

=

A(V0

,

V1

;

d) be 2-stage with Hom(V1

,

H*(M))

=

0.

Then £*(M)

=

1. D