FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 21

We denote this group by E#(M) for the remainder of the section. The minimal

model of

X, M

=

A(Vo, V1; d)

is 2-stage, where

Vo

=

(u1, ... ,

up,

v1, ... ,

vq) with

luj I

=

mj and lvk I

=

nk, and

Vi

= (

Wt, ... , Wq) with lwzl

=

2nt -1. The differential

is given by d(uj)

=

0

=

d(vk) and d(wt)

=

vf. A typical basis element of H*(M) is

represented by a cocycle of the form

u~ 1

• • •

u';vrj_I · · ·

v~q,

where each Ej and 'f/k is

either

0

or 1. We can express the degree of this cocycle using the following notation:

Let

m = (m1, ... ,

mp), n

=

(nt, ... , nq),

f.=

(El, ... , Ep) and

'f/ =

(rJt, ... , 'f/q)· Then

we set

u'v"~ = u~ 1

• • •

u';J'v7j_I · · ·

v~q

and, using the ordinary dot product, we have

that the degree of

u'v"~

is

f.·

m

+

rJ ·

n. We also write lEI

=

E

1

+ · · · +

f.p

and

1"11 = 'f/1

+ ... +

'f/q·

We wish to investigate the group E#(M). The elements of this group are

represented by maps f:

M

~

M.

By Corollary

3.4,

we may assume without loss

of generality that

f

IVa

=

t

IVa. We give a specialized version of the obstruction

theory of §§3 and 4 which is relevant to this situation. For each basis element

Wj

E

V1,

we have f(wj)

=

Wj +Xj· Since the cocycles

u'v"~

give a basis for H*(M),

we write

where the rationals

!3{.,.,

are determined uniquely and

~j E

M.

Here and in what

follows we adopt the convention that, in any equation which we write, the degrees

of elements that are added together and the degrees of elements that are equated

must agree. We further distinguish terms in the above sum and write for each

j

where again the rationals{.,., and

J.{.,.,

are determined uniquely by

f.

Now suppose that we have a second map f':

M

~

M

with[!']

E

f#(M)

and

!'Iva

=

tlva· We then represent

!'

as in

(7.1)

with coefficients -:~.,., and J.L:~.,., and

element

~j E

M.

7.2 Definition Let

J,

!' :

M

~

M be maps with [!], [!']

E

t'#(M) and !Iva

=

!'Iva

=

tlva· We write

f

and

!'

as in

(7.1)

and define the restricted obstruction

Ot(J,J')

E

Hom(V1,H*(M)) by

Ol(f,J')(wj)

= [

L

(.t.,.,-

.~~.,.,)u'v"~],

,,.,.,

'f/;=0

We denote this group by E#(M) for the remainder of the section. The minimal

model of

X, M

=

A(Vo, V1; d)

is 2-stage, where

Vo

=

(u1, ... ,

up,

v1, ... ,

vq) with

luj I

=

mj and lvk I

=

nk, and

Vi

= (

Wt, ... , Wq) with lwzl

=

2nt -1. The differential

is given by d(uj)

=

0

=

d(vk) and d(wt)

=

vf. A typical basis element of H*(M) is

represented by a cocycle of the form

u~ 1

• • •

u';vrj_I · · ·

v~q,

where each Ej and 'f/k is

either

0

or 1. We can express the degree of this cocycle using the following notation:

Let

m = (m1, ... ,

mp), n

=

(nt, ... , nq),

f.=

(El, ... , Ep) and

'f/ =

(rJt, ... , 'f/q)· Then

we set

u'v"~ = u~ 1

• • •

u';J'v7j_I · · ·

v~q

and, using the ordinary dot product, we have

that the degree of

u'v"~

is

f.·

m

+

rJ ·

n. We also write lEI

=

E

1

+ · · · +

f.p

and

1"11 = 'f/1

+ ... +

'f/q·

We wish to investigate the group E#(M). The elements of this group are

represented by maps f:

M

~

M.

By Corollary

3.4,

we may assume without loss

of generality that

f

IVa

=

t

IVa. We give a specialized version of the obstruction

theory of §§3 and 4 which is relevant to this situation. For each basis element

Wj

E

V1,

we have f(wj)

=

Wj +Xj· Since the cocycles

u'v"~

give a basis for H*(M),

we write

where the rationals

!3{.,.,

are determined uniquely and

~j E

M.

Here and in what

follows we adopt the convention that, in any equation which we write, the degrees

of elements that are added together and the degrees of elements that are equated

must agree. We further distinguish terms in the above sum and write for each

j

where again the rationals{.,., and

J.{.,.,

are determined uniquely by

f.

Now suppose that we have a second map f':

M

~

M

with[!']

E

f#(M)

and

!'Iva

=

tlva· We then represent

!'

as in

(7.1)

with coefficients -:~.,., and J.L:~.,., and

element

~j E

M.

7.2 Definition Let

J,

!' :

M

~

M be maps with [!], [!']

E

t'#(M) and !Iva

=

!'Iva

=

tlva· We write

f

and

!'

as in

(7.1)

and define the restricted obstruction

Ot(J,J')

E

Hom(V1,H*(M)) by

Ol(f,J')(wj)

= [

L

(.t.,.,-

.~~.,.,)u'v"~],

,,.,.,

'f/;=0