FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 15

which is nonzero. Thus by Proposition 4.7,

f..

and

f

11

are not homotopic. Therefore

{[f...]l ,\

E

Q*} is an infinite subset of

E*(M). D

Next we consider the subgroups

E#k(M)

of

E(M),

k:::;

oo, and give conditions

for these groups to be infinite. Notice that

E#k(M)

;;;:?

E#oo(M),

and so it suffices

to show

E#oo(M)

is infinite. Let

M

=

A(Vo,

V1

;

d).

In the sequel we adopt the

following notation: Let j : V0

--

H*(M)

be defined by j(v)

=

[v], let W denote

the cokernel of

j

and let

1r :

H* (M)

--

W

denote the projection. Also let

Vo

=

(v1, ... ,vr)

and

vl

=

(w1, ... ,ws)·

5.5 Proposition Let

M

=

A(V

0

,

V1

;

d)

be 2-stage. If Hom-

1

(V0

,

H*(M))

=

0 and

Hom(V1

,

W)

-=f.

0, then

E#oo(M)

is infinite.

Proof We proceed as in the previous proof, but now choose a cocycle

x E

M

such

that 1r[x]

i-

0. We show that

x

can be assumed to be decomposable. Write

r s

X=

LAjVj

+

LJ1kWk

+

x,

j=l k=1

where Aj, J1k

E

Q and X is decomposable. We apply d to obtain

Since M is in normal form and d: V1

--

A(V0

)

is injective, this yields

2::~=

1

J1kWk

=

0. Therefore

r

X=

LAjVj +X

j=l

and so

1r[x]

=

7r

[t,

AjVj]

+

7r[X]·

Thus 7r[X]

=

1r[x]

i-

0, and so we can replace

x

with

x

if necessary. We then proceed

as in the proof of Proposition 5.4 to define infinitely many homotopically distinct

maps

f.. :

M

--

M.

Since

with

X

decomposable,

[f...]

E

E#oo(M).

D

Finally, we turn to a consideration of the subgroup

E#n (M) E* (M)

n

E#n(M)

of

E(M).

Before establishing results we make an observation about this

subgroup that is relevant to many of our applications.

which is nonzero. Thus by Proposition 4.7,

f..

and

f

11

are not homotopic. Therefore

{[f...]l ,\

E

Q*} is an infinite subset of

E*(M). D

Next we consider the subgroups

E#k(M)

of

E(M),

k:::;

oo, and give conditions

for these groups to be infinite. Notice that

E#k(M)

;;;:?

E#oo(M),

and so it suffices

to show

E#oo(M)

is infinite. Let

M

=

A(Vo,

V1

;

d).

In the sequel we adopt the

following notation: Let j : V0

--

H*(M)

be defined by j(v)

=

[v], let W denote

the cokernel of

j

and let

1r :

H* (M)

--

W

denote the projection. Also let

Vo

=

(v1, ... ,vr)

and

vl

=

(w1, ... ,ws)·

5.5 Proposition Let

M

=

A(V

0

,

V1

;

d)

be 2-stage. If Hom-

1

(V0

,

H*(M))

=

0 and

Hom(V1

,

W)

-=f.

0, then

E#oo(M)

is infinite.

Proof We proceed as in the previous proof, but now choose a cocycle

x E

M

such

that 1r[x]

i-

0. We show that

x

can be assumed to be decomposable. Write

r s

X=

LAjVj

+

LJ1kWk

+

x,

j=l k=1

where Aj, J1k

E

Q and X is decomposable. We apply d to obtain

Since M is in normal form and d: V1

--

A(V0

)

is injective, this yields

2::~=

1

J1kWk

=

0. Therefore

r

X=

LAjVj +X

j=l

and so

1r[x]

=

7r

[t,

AjVj]

+

7r[X]·

Thus 7r[X]

=

1r[x]

i-

0, and so we can replace

x

with

x

if necessary. We then proceed

as in the proof of Proposition 5.4 to define infinitely many homotopically distinct

maps

f.. :

M

--

M.

Since

with

X

decomposable,

[f...]

E

E#oo(M).

D

Finally, we turn to a consideration of the subgroup

E#n (M) E* (M)

n

E#n(M)

of

E(M).

Before establishing results we make an observation about this

subgroup that is relevant to many of our applications.