10 MARTIN ARKOWITZ AND GREGORY LUPTON
homotopy type of a finite CW-complex,
My
is 2-stage and Hom(V1
,
H*(X;
Ql))
=
0
(e.g.,
X
andY can be spaces listed in (i) - (iii) above), then
[X, Y]-
Hom(H*(Y;
Ql),
H*(X;
Ql))
is finite-to-one. This is because the rationalization function [X, Y]
---
[XIQI, YIQI] is
finite-to-one [H-M-R,Cor.5.4]. In particular, for such spaces we have that if the
set of algebra homomorphisms
Hom(H*(Y;
Ql),
H*(X;
Ql))
is zero, then [X, Y] is a
finite set.
Section 4 - Obstruction Theory for Minimal Algebras, II
We further develop the obstruction theory of §3. Our goal is to prove a partial
converse to Proposition 3.3.
We show (Proposition 4.7) that, with some restrictions,
homotopic maps have zero obstruction. The notation of the previous section will
be used. We begin with two technical lemmas. Their proofs are straightforward,
and hence omitted.
4.1 Lemma Given a map
f:
M--- Nand a homomorphism 0: V1
---
H*(N),
there exists a map g: M--- N such that gjv0
=
fivo and 01(f,g)
=
0. 0
4.2 Lemma Given maps f,g, h: M--- N such that glvo
==
fivo
=
hiva· Then
0
We now prove another simple lemma.
4.3 Lemma If M
=
A(V0
,
V1
;
d) is 2-stage,
J,
g :
M
---
N
are maps such that
fivo
=
glvo and H : M
1
---
N is a homotopy from f tog, then Hlvo
=
0.
Proof If
Vj
is a basis element of V0
,
then
o:(
Vj)
=
Vj
+
Vj
by Lemma 3.2. Thus
Therefore H(vj)
=
0.
0
4.4 Proposition Let M
=
A(V
0
,
V1 ;d) be 2-stage and
f:,g:
M
---
N be maps
such that fivo
=
glvo. Let H : M
1
---
N be a homotopy starting at
f.
Then H
ends at g
{:::=::}
dHi(v)
+
Hid(v)
=
g(v)- f(v) for any v
E
Vo EB
V1.
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