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
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