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

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