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.

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.