FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 5

(ii) £*(X) is infinite

{:::::::

£* (XQ) is infinite and

(iii)

£*#(X) is infinite

{:::::::

£*#(XQ) is infinite.

Proof By Maruyama [Ma1] the natural homomorphism

e :

£#(X) ---+ £#(XQ)

obtained by rationalizing a homotopy class is the rationalization homomorphism.

Part (i) is an immediate consequence of this. Part (ii) follows similarly from [Ma2].

For

(iii)

we show that the natural homomorphism e' :£*#(X)---+ £*#(XQ) is a Q-

isomorphism. The kernel of e' is a torsion subgroup of £*#(X) since it is contained

in the kernel of

e :

£#(X) ---+ £#(XQ)· Next if 0

E

£*#(XQ) then we have to

show that there is an integer M such that

oM

E

Image e'. Clearly there exists

a

E

£#(X) and {3

E

£*(X) and integers m and n such that aQ

=

om and f3Q

=

on. Then (an )Q

=

({3m )Q

=

omn. Consider the sequence an

{3-m,

a2n{3-

2m, ...

of equivalences which all localize to the identity. Since the localization functor

[X,

X] ---+ [XQ, XQ] is finite-to-one [H-M-R,Cor.5.4], there exists a

k

such that

ak{3-k

=

1, and soak= {3k. Thus ak E £*#(X) and hence e'(ak)

=

(ak)Q

=

omk.

This establishes the result with M

=

mk. Thus e' : £*#(X) ---+ £*#(XQ) is a

Q-isomorphism and (iii) follows. 0

2.2 Remark In (iii) of Proposition 2.1 we showed that e' : £*#(X) ---+ £*# (XQ) is a

IQ-isomorphism. This is sufficient for our purposes. In fact, an additional argument

can be given to show that e' is the localization homomorphism.

2.3

Remark It is well-known [G-M,Chap.XIV] that there is a contravariant

equivalence between the homotopy category of rational spaces of finite type and the

homotopy category of finite type minimal algebras. From this it follows that if

M

is

the minimal model of a space

X

of dimension N, then there is an anti-isomorphism

between £(XQ) and £(M). Note that £*(XQ) is equal to the subgroup of £(XQ) of

homotopy equivalences which induce the identity on cohomology. Then £*(XQ) is

anti-isomorphic to £*(M). Furthermore, £#(XQ) is anti-isomorphic to £#N(M),

and so £*#(XQ) is isomorphic to £#N(M). In the sequel we study t'*(M), t'#N(M)

and t'#N(M) in order to obtain information about t'*(X), t'#(X) and t'*#(X) (see

Remark 5.9).

Section 3 - Obstruction Theory for Minimal Algebras, I

In this section and the next we develop a simple obstruction theory for ho-

motopy of maps of a 2-stage minimal algebra into an arbitrary minimal algebra.

(ii) £*(X) is infinite

{:::::::

£* (XQ) is infinite and

(iii)

£*#(X) is infinite

{:::::::

£*#(XQ) is infinite.

Proof By Maruyama [Ma1] the natural homomorphism

e :

£#(X) ---+ £#(XQ)

obtained by rationalizing a homotopy class is the rationalization homomorphism.

Part (i) is an immediate consequence of this. Part (ii) follows similarly from [Ma2].

For

(iii)

we show that the natural homomorphism e' :£*#(X)---+ £*#(XQ) is a Q-

isomorphism. The kernel of e' is a torsion subgroup of £*#(X) since it is contained

in the kernel of

e :

£#(X) ---+ £#(XQ)· Next if 0

E

£*#(XQ) then we have to

show that there is an integer M such that

oM

E

Image e'. Clearly there exists

a

E

£#(X) and {3

E

£*(X) and integers m and n such that aQ

=

om and f3Q

=

on. Then (an )Q

=

({3m )Q

=

omn. Consider the sequence an

{3-m,

a2n{3-

2m, ...

of equivalences which all localize to the identity. Since the localization functor

[X,

X] ---+ [XQ, XQ] is finite-to-one [H-M-R,Cor.5.4], there exists a

k

such that

ak{3-k

=

1, and soak= {3k. Thus ak E £*#(X) and hence e'(ak)

=

(ak)Q

=

omk.

This establishes the result with M

=

mk. Thus e' : £*#(X) ---+ £*#(XQ) is a

Q-isomorphism and (iii) follows. 0

2.2 Remark In (iii) of Proposition 2.1 we showed that e' : £*#(X) ---+ £*# (XQ) is a

IQ-isomorphism. This is sufficient for our purposes. In fact, an additional argument

can be given to show that e' is the localization homomorphism.

2.3

Remark It is well-known [G-M,Chap.XIV] that there is a contravariant

equivalence between the homotopy category of rational spaces of finite type and the

homotopy category of finite type minimal algebras. From this it follows that if

M

is

the minimal model of a space

X

of dimension N, then there is an anti-isomorphism

between £(XQ) and £(M). Note that £*(XQ) is equal to the subgroup of £(XQ) of

homotopy equivalences which induce the identity on cohomology. Then £*(XQ) is

anti-isomorphic to £*(M). Furthermore, £#(XQ) is anti-isomorphic to £#N(M),

and so £*#(XQ) is isomorphic to £#N(M). In the sequel we study t'*(M), t'#N(M)

and t'#N(M) in order to obtain information about t'*(X), t'#(X) and t'*#(X) (see

Remark 5.9).

Section 3 - Obstruction Theory for Minimal Algebras, I

In this section and the next we develop a simple obstruction theory for ho-

motopy of maps of a 2-stage minimal algebra into an arbitrary minimal algebra.