A criterion of \(SNT(X) = \{[X]\}\) for hyperformal spaces (Q1040201)

For a positive integer \(n\), let \(X^{(n)}\) denote the Postnikov section of a space \(X\) through dimension \(n\), and we say that the spaces \(X\) and \(Y\) have the same \(n\)-type if \(X^{(n)}\) and \(Y^{(n)}\) are homotopy equivalent. Let \(SNT(X)\) denote the set of homotopy types of spaces \(Y\) having the \(n\)-types of \(X\) for all \(n\), and let \(\Aut(X)\) denote the group of homotopy classes of self-homotopy equivalences of \(X\). In this paper, the author studies the set \(SNT(X)\) for a simply connected formal space \(X\) with finite type such that there exists some number \(N>\max\{n:\pi_n(X)\otimes \mathbb Q \not= 0\}\). In particular, as a generalization of a result due to \textit{C. A. McGibbon} and \textit{J. M. Møller} [Topology 31, No. 1, 177--201 (1992; Zbl 0765.55010)], he proves that if the natural projection map \(\Phi:\Aut(X^{(N)})\to \Aut H^*(X^{(N)},\mathbb Z)\) has finite cokernel, the condition \(SNT(X)=\{[X]\}\) is equivalent to one of the following conditions: {\parindent=6mm \begin{itemize}\item[(1)] The canonical map \(\Aut(X)\to \Aut(X^{(n)})\) has finite cokernel for all \(n>N\). \item[(2)] \(\Phi :\Aut(X)\to \Aut H^{\leq n}(X^{(n)},\mathbb Z)\) has finite cokernel for all \(n>N\). \end{itemize}}