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 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
Previous Page Next Page