On the same \(N\)-type structure for the suspension of the Eilenberg-Mac Lane spaces
For a connected space \(X\), let \(\text{Aut}\;X\) denote the group of homotopy classes of self-homotopy equivalences over \(X\), and let \(SNT(X)\) be the set of all homotopy types \([Y]\) such that the Postnikov approximations \(X^{(n)}\) and \(Y^{(n)}\) are homotopy equivalent for each \(n\geq 1\). It is said that a homomorphism \(\varphi :G\to H\) between groups has finite cokernel if the image \(\varphi (G)\) has finite index in \(H\). If \(X\) is a simply connected space with finite type and it has the rational homotopy type of a bouquet of spheres, it is known that \(SNT(X)=*\) if the natural map \(\phi_n:\text{Aut}\;X\to \text{Aut}(\pi_{\leq n}X)\) given by \(\phi_n (f)=\pi_{\leq n}(f)\) has finite cokernel for each \(n\) [\textit{C.A. McGibbon} and \textit{J. Møller}, Algebraic topology, Proc. Conf., S. Feliu de Guixols/Spain 1990, Lect. Notes Math. 1509, 285--293 (1992; Zbl 0768.55008)]. In this paper, by using the above result, the author proves that \(SNT(\Sigma K(\mathbb Z,2m))=*\) for all \(m\geq 1\). Because the case \(m=1\) was already proved in [\textit{D.-W. Lee}, Proc. Am. Math. Soc. 137, No.~3, 1161--1168 (2009; Zbl 1162.55006)], he treats the case \(m\geq 2\). To prove this he constructs a family of maps \(\{\hat{\psi}_n:K\to \Omega \Sigma K,\hat{x}_n:S^{2mn}\to \Omega \Sigma K\}_{n\geq 1}\) by using a technique due to [\textit{K. Morisugi}, J. Math. Kyoto Univ. 38, No.~1, 151--165 (1998; Zbl 0924.55002)] and he shows that there are sufficiently many self maps over \(\Sigma K\), where \(K=K(\mathbb Z,2m)\).
the group of homotopy classes of self-homotopy equivalences
scientific article; zbMATH DE number 5726608