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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