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
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