8 MARTIN ARKOWITZ AND GREGORY LUPTON
for some
Zk
EN,
k
=
1, ... , s.
Define H: M
1
--t
Non
generators by
and H
=
0 on all other generators of M
1
=
A(V EB V EB
V).
Then His a DG algebra
homomorphism. We set g'
=
H
a/M :
M
--t
N,
and have
l
~
g. By Lemma 3.2,
and
g'(vj)
=
Ha(vj)
g'(wk)
=
Ha(wk)
=
H(vj)
+
H(vj)
=
g(vj)
=
H(wk)
+
H(wk)
+
H(yk)
=
g(wk)
+
d(zk)
=
f(wk)·
with
Yk
E
(Vo)
D
3.4 Corollary If
M
=
A(V
0
,
V1;
d)
and /, g :
M
--t
N
are maps such that
f*
=
g* : H*(M)
--t
H*(N),
then there exists a map
g' : M
--t
N
such that
g'
~
g
and
g'/vo
=
f/Vo·
Proof We show that the obstruction
Oo(f,g) : Vo
--t
H*(N)
is zero. If
v
E
VQ,
then
Oo(f,g)(v)
=
[f(v)- g(v)]
=
f*[v]- g*[v/
=
0. D
Next let I: [M,N]
--t
Hom(H*(M),
H*(N))
be the function which assigns to
a homotopy class the induced cohomology homomorphism.
3.5 Corollary If
M
=
A(V
0
,
V1; d) is a 2-stage minimal algebra,
N
is any minimal
algebra and Hom(V1,
H*
(N))
=
0, then
I:
[M,N]
--t
Hom(H*(M),H*(N))
is one-one.
Proof This follows from Corollary 3.4 and Proposition 3.3 (ii).
D
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