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

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