FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 7

Vo

and

{WI' . ..

'Ws}

of

vl.

Recall from §2 the DG algebra

M

1

=

A(V

EB

v

EB

Y)

and the maps a, 'Y :

M

1

----+

M

1

.

3.2 Lemma

With the above notation, we have

(i) 'Y(v1)

=

v

1

and a(v1)

=

Vj

+

Vj·

(ii) (a) 'Y(

w1)

= w

1

+

x1, where x1 is in the ideal

(V o),

and

a( w1)

=

w1

+

Wj

+

YJ,

where

YJ

E

(Vo).

(b)

"f 2 (wJ)

is in the ideal

(Yo)

and

a(wJ)

=

Wj

+

"f(wj)

+

Zj,

where

ZJ

E

(Y0

).

Proof Straightforward, hence omitted. D

Now we introduce an obstruction theory for homotopy of maps of M into an

arbitrary minimal algebra

N.

We consider two cases : (i)

J,

g : M

----+

N

are

arbitrary maps, and (ii)

J,

g :

M

----+

N

satisfy

!Iva = glva.

In the first case we

define an obstruction 0

0

(!,g) : V0

----+

H* (N) and in the second case we define an

obstruction 0 1(/,g): V1

----+

H*(N). To handle both cases at once we set V_

1 =

0

and suppose that

!I

- gl

either for l

=

0 or for l

=

1.

Vi-1 - Vi-1'

If

v E

\!i,

then

d(f(v)- g(v))

=

f(d(v))- g(d(v))

=

0,

since d(v) E A(V'i-1) and

fiVt-

1

=

gl1/i_

1

•

Thus f(v)- g(v) is a cocycle inN. We

then define an obstruction Oz (!,g) :

V'i

----+

H* (N) by

Oz(f,g)(v)

=

[f(v)- g(v)]

E

H*(N).

Thus there are two possible obstructions

(i) Oo(f,g): Vo----+ H*(N) for arbitrary maps f and g and

(ii) 01(f,g):

Vi----+

H*(N) in the case

!Iva= glva·

3.3 Proposition Let

M

=

A(V

0

,

V1;

d)

be 2-stage and

J,

g :

M ___. N

be maps.

(i)

If

Oo(f,g)

=

0 : Vo

----+

H*(N), then there is a map g' : M

----+

N such that

g'

~

g and

g'lva = !Iva·

(ii)

If

!Iva= glva

and Ol(f,g)

=

0:

vl ___.

H*(N), then f

~g.

Proof We only prove case (ii) because case (i) is analogous. We have a basis

{w1, ... ,ws} ofV1 and

Vo

and

{WI' . ..

'Ws}

of

vl.

Recall from §2 the DG algebra

M

1

=

A(V

EB

v

EB

Y)

and the maps a, 'Y :

M

1

----+

M

1

.

3.2 Lemma

With the above notation, we have

(i) 'Y(v1)

=

v

1

and a(v1)

=

Vj

+

Vj·

(ii) (a) 'Y(

w1)

= w

1

+

x1, where x1 is in the ideal

(V o),

and

a( w1)

=

w1

+

Wj

+

YJ,

where

YJ

E

(Vo).

(b)

"f 2 (wJ)

is in the ideal

(Yo)

and

a(wJ)

=

Wj

+

"f(wj)

+

Zj,

where

ZJ

E

(Y0

).

Proof Straightforward, hence omitted. D

Now we introduce an obstruction theory for homotopy of maps of M into an

arbitrary minimal algebra

N.

We consider two cases : (i)

J,

g : M

----+

N

are

arbitrary maps, and (ii)

J,

g :

M

----+

N

satisfy

!Iva = glva.

In the first case we

define an obstruction 0

0

(!,g) : V0

----+

H* (N) and in the second case we define an

obstruction 0 1(/,g): V1

----+

H*(N). To handle both cases at once we set V_

1 =

0

and suppose that

!I

- gl

either for l

=

0 or for l

=

1.

Vi-1 - Vi-1'

If

v E

\!i,

then

d(f(v)- g(v))

=

f(d(v))- g(d(v))

=

0,

since d(v) E A(V'i-1) and

fiVt-

1

=

gl1/i_

1

•

Thus f(v)- g(v) is a cocycle inN. We

then define an obstruction Oz (!,g) :

V'i

----+

H* (N) by

Oz(f,g)(v)

=

[f(v)- g(v)]

E

H*(N).

Thus there are two possible obstructions

(i) Oo(f,g): Vo----+ H*(N) for arbitrary maps f and g and

(ii) 01(f,g):

Vi----+

H*(N) in the case

!Iva= glva·

3.3 Proposition Let

M

=

A(V

0

,

V1;

d)

be 2-stage and

J,

g :

M ___. N

be maps.

(i)

If

Oo(f,g)

=

0 : Vo

----+

H*(N), then there is a map g' : M

----+

N such that

g'

~

g and

g'lva = !Iva·

(ii)

If

!Iva= glva

and Ol(f,g)

=

0:

vl ___.

H*(N), then f

~g.

Proof We only prove case (ii) because case (i) is analogous. We have a basis

{w1, ... ,ws} ofV1 and