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.
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