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.
Previous Page Next Page