4 MARTIN ARKOWITZ AND GREGORY LUPTON
A homotopy from M
toN
is a map H : M
1
---+
N.
Then His a homotopy
from f tog if
HIM
= f and
Ho:IM
=g. We say that
H
begins at f and ends at g,
and we write
f
~g.
We denote the collection of homotopy classes by [M,Af] with
the homotopy class of a map
f : M
---+
N
written
[!].
The group of homotopy classes of homotopy equivalences of a minimal alge-
bra
M
is denoted
£(M).
In §5 we study the following subgroups of
£(M):
(i)
£*(M),
the subgroup of homotopy classes which induce the identity on
H*(M),
(ii)
£#k(M),
the subgroup of homotopy classes which induce the identity on the
indecomposables
Qi(M)
of
M,
for all
i
:S:
k, where k
~;
oo, (iii)
£#k(M)
=
£*(M)
n
£#k(.M).
We say that a topological space is of finite type if it has rational cohomology
of finite dimension in each degree. All topological spaces in this paper are based
and have the based homotopy type of a nilpotent CW-complex of finite type. For
a space X, we denote by XIQI the rationalization of X and for a map
J,
we denote
by
fiQI
the rationalization of
f.
We denote the Sullivan minimal model of a space
X by Mx [G-M].
Now let X have the homotopy type of a finite complex of dimension N, or of
the rationalization of such a finite complex, and define £*(X) to be the kernel of
the homomorphism
£(X)---+
L
AutHi(X)
i~N
and £#(X) to be the kernel of the homomorphism
£(X)---+
L
Aut?T;(X).
i~N
Then Dror and Zabrodsky have shown that £*(X) and £#(X) are nilpotent groups
[D-Z],
and it follows that £*#(X)= £*(X)
n
£#(X) is also nilpotent. Thus these
groups can be rationalized [H-M-R]. The following proposition relates the sub-
groups of £(X) with those of £(XIQI)·
2.1 Proposition Let X have the homotopy type of a finite, simple CW-complex.
Then
(i) £#(X) is infinite
-===?-
£#(XIQI) is infinite.
Assume further that
X
is 1-connected. Then
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