6

MARTIN ARKOWITZ AND GREGORY LUPTON

Although it is clear that this theory can be generalized in different ways (e.g., to

n-stage minimal algebras), we have tailored the exposition to fit the applications

we give in §§5, 6 and 7 and also in a subsequent paper [Ar-Lu].

We begin by discussing some of our notation and conventions. A minimal DG

algebra (A(V), d) is said to be in normal form if the following condition holds :

If

v

E

V and d(v)

=

d(x) for some decomposable element x, then d(v)

=

0. It is

known that every minimal DG algebra can be assumed to be in normal form (

cf.

[Pallp.172]). This is easily seen as follows: Define a graded subspace W of V by

v

E

W

~

d(v)

=

d(x) for some decomposable X· Then V

=

W

E9

W',

where

W'

is a complementary subspace to W. Choose a basis w1, ... , Wr for W and let U be

the subspace of A(V) given by U

=

(w1- Xl, ... , Wr- Xr), where d(wi)

=

d(Xi) for

some Xi· Then clearly A(U E9 W')

=

A(V). We then verify that A(U E9 W') is in

normal form:

If

v E U E9

W'

and d(v)

=

d(x) for some decomposable x, then write

v

=

u

+

w', for u E U and w' E W'. Then d(w')

=

d(u)

+

d(w')

=

d(v)

=

d(x),

and sow' E

W.

Thus w'

=

0, and therefore d(v)

=

d(u)

==

0. This completes the

argument.

Let (M, d) be a minimal DG algebra. We call M a 2-stage DG algebra

if

M

=

A(Vo E9 V1), for graded vector spaces Vo and V1 with d(Vo)

=

0 and dlv1

V1 ___. A(V0

).

We denote this by M

=

(M,d)

=

A(V

0

,

V1;d).

We note that for a 2-stage minimal algebra A(V

0

,

V1;d) there may be various

choices of subspaces V0 and V1. However, one can always choose a decomposition

in which d: V1 ___. A(V

0

)

is injective.

For the remainder of the paper we assume that all minimal DG algebras are in

normal form and furthermore that all 2-stage algebras A(Vo, V1; d) have d : V1

---*

A(Vo) injective.

3.1 Remark

If X

is a 1-connected space of finite type with 2-stage minimal model

A(V

0

,

V1;d) and if hn : 1T'n(X)

®

Q---*

Hn(X;Q) is the rational Hurewicz homo-

morphism, then it can be shown that Image hn is isomorphic to V0n and that Ker

hn is isomorphic to V1 n. In the sequel we give results with hypotheses phrased in

terms of V0 and V1 (see Remarks 3.7, Corollary 4.9 and Remark 5.9). Clearly such

hypotheses could be replaced with purely topological hypotheses.

Now let

(M, d)

=

A(V

0

,

V1;

d)

be 2-stage and let us fix a basis { v1, ... ,

Vr}

of

MARTIN ARKOWITZ AND GREGORY LUPTON

Although it is clear that this theory can be generalized in different ways (e.g., to

n-stage minimal algebras), we have tailored the exposition to fit the applications

we give in §§5, 6 and 7 and also in a subsequent paper [Ar-Lu].

We begin by discussing some of our notation and conventions. A minimal DG

algebra (A(V), d) is said to be in normal form if the following condition holds :

If

v

E

V and d(v)

=

d(x) for some decomposable element x, then d(v)

=

0. It is

known that every minimal DG algebra can be assumed to be in normal form (

cf.

[Pallp.172]). This is easily seen as follows: Define a graded subspace W of V by

v

E

W

~

d(v)

=

d(x) for some decomposable X· Then V

=

W

E9

W',

where

W'

is a complementary subspace to W. Choose a basis w1, ... , Wr for W and let U be

the subspace of A(V) given by U

=

(w1- Xl, ... , Wr- Xr), where d(wi)

=

d(Xi) for

some Xi· Then clearly A(U E9 W')

=

A(V). We then verify that A(U E9 W') is in

normal form:

If

v E U E9

W'

and d(v)

=

d(x) for some decomposable x, then write

v

=

u

+

w', for u E U and w' E W'. Then d(w')

=

d(u)

+

d(w')

=

d(v)

=

d(x),

and sow' E

W.

Thus w'

=

0, and therefore d(v)

=

d(u)

==

0. This completes the

argument.

Let (M, d) be a minimal DG algebra. We call M a 2-stage DG algebra

if

M

=

A(Vo E9 V1), for graded vector spaces Vo and V1 with d(Vo)

=

0 and dlv1

V1 ___. A(V0

).

We denote this by M

=

(M,d)

=

A(V

0

,

V1;d).

We note that for a 2-stage minimal algebra A(V

0

,

V1;d) there may be various

choices of subspaces V0 and V1. However, one can always choose a decomposition

in which d: V1 ___. A(V

0

)

is injective.

For the remainder of the paper we assume that all minimal DG algebras are in

normal form and furthermore that all 2-stage algebras A(Vo, V1; d) have d : V1

---*

A(Vo) injective.

3.1 Remark

If X

is a 1-connected space of finite type with 2-stage minimal model

A(V

0

,

V1;d) and if hn : 1T'n(X)

®

Q---*

Hn(X;Q) is the rational Hurewicz homo-

morphism, then it can be shown that Image hn is isomorphic to V0n and that Ker

hn is isomorphic to V1 n. In the sequel we give results with hypotheses phrased in

terms of V0 and V1 (see Remarks 3.7, Corollary 4.9 and Remark 5.9). Clearly such

hypotheses could be replaced with purely topological hypotheses.

Now let

(M, d)

=

A(V

0

,

V1;

d)

be 2-stage and let us fix a basis { v1, ... ,

Vr}

of