Classification of spaces of the same $n$-type for all $n$

Abstract

The set of homotopy equivalence classes of CW-spaces Y for which the Postnikov approximations are homotopy equivalent to those of a given space X is shown to be in one-to-one correspondence with liml of a tower of homotopy automorphism groups. Recall that two CW-spaces X and Y are said to have the same n-type if the nth Postnikov approximations X(n) and y(n) are homotopy equivalent. The homotopy classification problem for infinite dimensional complexes can be studied in two steps. First, classify all finite dimensional complexes (!) and then classify for a given X all complexes of the same n-type for all n as X. The purpose of this note is to codify this second step. THEOREM I. Let X be a connected CW-space. Denote by SNT(X) the set of homotopy equivalence classes of CW-spaces Y such that Y has the same n-type for all n as X. Then SNT(X) is in one-to-one correspondence with the pointed set lim1(Aut(X (n))), where Aut(X (n)) is the group of homotopy classes of homotopy equivalences of X(n).