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.
Previous Page Next Page