On spaces with the same \(n\)-type for all \(n\)
Let \(X^{(n)}\) be the \(n\)-th Postnikov section of a space \(X\). Let \(SNT(X)\) be the set of homotopy types of spaces \(Y\) such that \(X^{(n)}\) and \(Y^{(n)}\) are homotopy equivalent for all \(n\). In a paper of \textit{C. Wilkerson} [Proc. Am. Math. Soc. 60(1976), 279-285 (1977; Zbl 0345.55010)] it is proved that \(SNT(X)\approx \lim^ 1\Aut(X^{(n)})\) for a connected CW-complex \(X\); here Aut denotes the group of homotopy classes of self-homotopy equivalences. The authors prove the following purely algebraic result: If \((G_ n)\) is an inverse system of countable groups, then either this system satisfies the Mittag-Leffler condition (whence \(\lim^ 1\) is trivial) or \(\lim^ 1 G_ n\) is uncountable. Thus if \(X\) is a nilpotent space of finite type, then \(SNT(X)\) is either uncountably large or trivial. Particularly interesting are the cases \(X=BG\) for a compact connected Lie group \(G\). Here the result is: \(SNT(BG)\) is trivial if and only if \(G\) is a torus or \(G=SU(n)\) or \(=PSU(n)\) and \(n=2,3\). (On the other hand, it is shown that \(SNT(BU)\) and \(SNT(BSp)\) are trivial.) Another interesting case is \(X=\Omega K\) where \(K\) is a 1-connected finite CW-complex. The authors conjecture that \(SNT(X)\) is trivial in this case. This is proven to be true if \(K\) is an \(H_ 0\)-space.
scientific article; zbMATH DE number 54637