FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 3

infinite in terms of the dimensions of the spheres.

Section 2- Minimal Algebras and Rationalization

We begin with some algebraic preliminaries. In general we adopt the conven-

tions of [G-M], [H-S] and [D-G-M-S]. A collection V

=

{Vk

I

k an integer?: 0},

where each Vk is a vector space over the rationals

Q,

is called a graded vector

space. We write v

E

V to indicate that v

E

Vk for some k and let

lvl

denote the

degree

k

of

v.

We shall assume wherever convenient that

V

is finite dimensional.

If v1, ... , Vr is a basis of V, then we write V

= (

v1, ... , Vr).

Let

A

be a differential graded commutative algebra (DG algebra, for short). In

this paper, we only consider DG algebras that have cohomology of finite type, i.e.,

that have cohomology which is finite dimensional in each degree. For a cocycle z

E

A we let [ z]

E

H* (A) denote the cohomology class of z. By a map

f :

A

--+

B of DG

algebras is meant a DG algebra homomorphism. The identity map of

A

will always

be denoted by

~

:

A

--+

A.

If

A

is the free graded commutative algebra generated

by the graded vector space V, then we write

A

=

A(V). If V

= (

v1, ... , Vr), then

A=

A(v1, ... , Vr)ยท

A DG algebra M is called a minimal algebra if (i) M

=

A(V) for some

vector space V and ( ii) there is a basis v1, ... , Vr, . . . for V such that d( Vr)

E

A(v1

, ...

,vr_

1),

where dis the differential of

M

[G-M]. We denote by

(S)

the

ideal of

M

generated by a subset

S

of

M.

For maps

J,

g :

M

--+

N

of minimal algebras we use the notion of homotopy

given in [H-S,p.240] which we now describe. Suppose M

=

A(V) with differential

d and define a DG algebra M

1

=

A(V EB V EB

V)

with differential also called d

as follows :

V

is an isomorphic copy of V and V is the desuspension of V (i.e.,

vp

=

vp+l ). Furthermore, the differential

d

of

M

1

agrees with the differential on

M, d(v)

=

v and d(v)

=

0, for v

E

V and v

E

V.

In addition, there is a degree

-1 derivation i: M

1

--+

M

1

defined by i(v)

=

v, i(v)

=

0 and i(v)

=

0. We then

obtain a degree 0 derivation

'Y :

M

1

--+

M

1

by setting

'Y

=

di

+

id

=

[d,

i]

(the

bracket of graded derivations). Finally, we have a map

a:

M

1

--+

M

1

defined by

00

1

-'""' n

a-

~~'Y.

n=O

n.

infinite in terms of the dimensions of the spheres.

Section 2- Minimal Algebras and Rationalization

We begin with some algebraic preliminaries. In general we adopt the conven-

tions of [G-M], [H-S] and [D-G-M-S]. A collection V

=

{Vk

I

k an integer?: 0},

where each Vk is a vector space over the rationals

Q,

is called a graded vector

space. We write v

E

V to indicate that v

E

Vk for some k and let

lvl

denote the

degree

k

of

v.

We shall assume wherever convenient that

V

is finite dimensional.

If v1, ... , Vr is a basis of V, then we write V

= (

v1, ... , Vr).

Let

A

be a differential graded commutative algebra (DG algebra, for short). In

this paper, we only consider DG algebras that have cohomology of finite type, i.e.,

that have cohomology which is finite dimensional in each degree. For a cocycle z

E

A we let [ z]

E

H* (A) denote the cohomology class of z. By a map

f :

A

--+

B of DG

algebras is meant a DG algebra homomorphism. The identity map of

A

will always

be denoted by

~

:

A

--+

A.

If

A

is the free graded commutative algebra generated

by the graded vector space V, then we write

A

=

A(V). If V

= (

v1, ... , Vr), then

A=

A(v1, ... , Vr)ยท

A DG algebra M is called a minimal algebra if (i) M

=

A(V) for some

vector space V and ( ii) there is a basis v1, ... , Vr, . . . for V such that d( Vr)

E

A(v1

, ...

,vr_

1),

where dis the differential of

M

[G-M]. We denote by

(S)

the

ideal of

M

generated by a subset

S

of

M.

For maps

J,

g :

M

--+

N

of minimal algebras we use the notion of homotopy

given in [H-S,p.240] which we now describe. Suppose M

=

A(V) with differential

d and define a DG algebra M

1

=

A(V EB V EB

V)

with differential also called d

as follows :

V

is an isomorphic copy of V and V is the desuspension of V (i.e.,

vp

=

vp+l ). Furthermore, the differential

d

of

M

1

agrees with the differential on

M, d(v)

=

v and d(v)

=

0, for v

E

V and v

E

V.

In addition, there is a degree

-1 derivation i: M

1

--+

M

1

defined by i(v)

=

v, i(v)

=

0 and i(v)

=

0. We then

obtain a degree 0 derivation

'Y :

M

1

--+

M

1

by setting

'Y

=

di

+

id

=

[d,

i]

(the

bracket of graded derivations). Finally, we have a map

a:

M

1

--+

M

1

defined by

00

1

-'""' n

a-

~~'Y.

n=O

n.