12
MARTIN ARKOWITZ AND GREGORY LUPTON
generated by
W4n-1·
Define
f.. :
M ----. M
to be the identity on
V2n-l
and
v2n
and
f..(w4n-d
=
W4n-1
+
AV2n-1V2n,
with
A
=I
0. Then
!.Iva
=~Iva
and thus
OI(f..,
~):Vi_____.
H*(M)
is defined. Note 01(f..,
~)(w4n-1)
=
[f(w4n-l)
-W4n-1]
=
.X[v2n-1V2n]
=I
0, so
01(f..,~)
=I
0. But we now show
f..
~~by
constructing a
homotopy H:
M
1
----.
M.
Let
H
start at
f..
and set
H(ihn)
=
-~V2n-l·
Let
H
be
zero on the other generators. Clearly H is a homomorphism of DG algebras, and
by Proposition 4.4,
H
ends at
t.
Therefore
H
is a homotopy from
f..
to
~.
Thus an additional hypothesis is needed for the converse of Proposition 3.3.
We denote by Hom-
1(Vo,H*(N))
the degree
-1
homomorphisms of the graded
vector space
Vo
into the graded vector space
H*
(N).
4.7 Proposition Let
M
be 2-stage,
J,g: M ----. N
maps such that
!Iva
=
glva
and Hom-
1(Vo,H*(N))
=
0.
Iff~
g,
then
0 1(f,g)
=
0: V1
----.
H*(N).
Proof By Corollary 4.5,
01(f,g)(wk)
=
-[Hidwk],
where His a homotopy from
f
to
g.
Thus it suffices to show that
Hid(wk)
is a coboundary. Now
d(wk)
E
A(V
0
),
so
d( wk)
is a linear combination of terms of the form
Vj
1

Vj,,
where { v1, ... ,
Vr}
is a basis of
V0
and
t
~
2. Hence
id(wk)
is a linear combination of terms of the form
Vj
1
...
fjJk · "Vj,·
We show that
H(vj
1
..
·Vjk ... vj,)
=
H(vjJ .. ·H(vjk) .. ·H(vj,)
is a coboundary. First observe that
dH(vj)
=
Hd(vj)
=
0 by Lemma 4.3. Thus
O(vj)
=
[H(vj)] defines a homomorphism():
V0
----.
H*(N)
of degree
-1.
By hy-
pothesis, ()
=
0. Therefore, for every
j
=
1, ... , k, H(
Vj)
is a co boundary. Hence
in the expression
H(vjJ · · · H(vjk) · · · H(vj,)
the element
H(vjk)
is a coboundary
and the elements
H(vj
1
), ...
,H(vjk_J,H(vjk+J, ... ,H(vj,)
are cocycles. Thus
H(vjJ · · ·H(vjk) · · ·H(vj,)
is a coboundary. This completes the proof. D
This result can be used to give conditions under which [M,J\f] is infinite.
4.8 Proposition Let
M
=
A(V
0
,
V1
;
d)
be a 2-stage minimal algebra and assume
that Hom-
1
(V0
,
H*(N))
=
0 and that Hom(V1
,
H*(N))
=I
0. Then the set [M,J\f]
is infinite.
Proof Since Hom(V1,
H*
(N))
=I
0, there exist infinitely many distinct homomor-
phisms
()i :
V1
----.
H*(N).
Fix any map
f : M ----.
Nand apply Lemma 4.1 to
obtain maps
g; : M----. N
such that
gilva =!Iva
and
01(f,gi)
=
Oi.
It
suffices to
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