On the generalized same \(N\)-type conjecture (Q2927888)

Let \(X\) be a connected CW-complex. This article concerns the set of same \(N\)-types of \(X\), \(SNT(X)\). This is the set of all homotopy types \([X']\) such that the \(n\)th Postnikov approximations \(X^{(n)}\) and \(X{}'{}^{(n)}\) are homotopy equivalent for all \(n\). When \(X\) is finite-dimensional or has only finitely may non-zero homotopy groups, \(SNT(X) = *\)NEWLINENEWLINEThe main theorem shows \(SNT(\Sigma Y) = *\) for any space \(Y\) of the form NEWLINE\[NEWLINE Y = K(\mathbb{Z}, 2a) \wedge \bigwedge_{b = 1}^{2k} K(\mathbb{Z}, 2b + 1) NEWLINE\]NEWLINE where \(a\) and \(k\) are positive integers. This is an extension of the author's previous work showing \(SNT(\Sigma K(\mathbb{Z},2a))\) is trivial: [Proc. Am. Math. Soc. 137, No. 3, 1161--1168 (2009; Zbl 1162.55006)] and [J. Pure Appl. Algebra 214, No. 11, 2027--2032 (2010; Zbl 1196.55007)]NEWLINENEWLINEThe method of proof uses the following theorem of \textit{C. A. McGibbon} and \textit{J. M. Møller} [Topology 31, No. 1, 177--201 (1992; Zbl 0765.55010)]: If \(X\) is a 1-connected space with finite type over a subring of the rationals and has the rational homotopy type of a boquet of spheres, then \(SNT(X) = *\) if and only if the map \(\mathrm{Aut}(X) \rightarrow \mathrm{Aut}(\pi_{\leq n}(X))\) has finite cokernel for all \(n\).NEWLINENEWLINEThe application of this theorem depends on a careful study of self maps of \(\Sigma Y\), and this occupies the majority of the paper. The author uses an inductive argument based on a filtration of Whitehead products which may be of independent interest.