Spaces of the same clone type (Q1363221)

Two spaces have the same clone type when they are STN-equivalent (the \(n\)-stage Postnikov towers \(X^{(n)}\) and \(Y^{(n)}\) are homotopy equivalent for all \(n)\) and have the same genus \((p\)-localizations are homotopy equivalent for all prime \(p)\). In [Comment. Math. Helv. 68, No. 2, 263-277 (1993; Zbl 0799.55007)] \textit{C. A. McGibbon} showed that the family of spaces \(\{N^1(I,J) =M^1(I,J) \times M^1(J,I) \mid\{I,J\}\) partition of all primes\} (where \(M^n(I,J)\) is a pullback of \(S^{2m+ 1}_I \to K(Q,2m+1) \leftarrow K(Z_J,2m+1))\) has the same clone type and classified the homotopy type by using the ordinary cohomology operations. In this paper the author calculates \(\text{End} (\Omega^k N^m(I,J))\) and classifies the homotopy type by using \(\text{End} (-)\). He also discriminates between \(N^1(I,J)\) and the example \(A\) of [\textit{C. A. McGibbon} and \textit{J. M. Møller}, Lond. Math. Soc. Lect. Note Ser. 175, 131-143 (1992; Zbl 0758.55002)]. In section 2, theorem 2.1, for \(2m>k>0\) he shows that the fiber spaces \(E(f)\) induced by \(f:K(Z,2m-k+1) \to\Omega^{k-1} S^{2m+1}\) are not homotopy equivalent to \(\Omega^k N^m(I,J)\) for nontrivial partition \(\{I,J\}\). In [Yokohama Math. J. 41, No. 1, 17-24 (1993; Zbl 0792.55003); Math. J. Okayama Univ. 34, 217-223 (1992; Zbl 0860.55009)] the author classified the homotopy type of \(\{\Omega^k C(f) \mid f:\Sigma^k CP^\infty \to S^{k+3}\}\) for \(k=0,1,2, \dots, \infty\). Now he shows that for maps \(\{f\}\) in the kernel of local expansion \(\text{Ph} (\Sigma^k CP^\infty, S^{k+3}) \to\text{Ph} (\Sigma^k CP^\infty, \Pi_p (S^{k + 3})_{(p)})\) he gets uncountably many infinite loop spaces of the same clone type. In the last section he calculates the set \(\text{Ph} (\Omega^k M^m(I,J) \Omega^hS^t)\) of phantom maps generalizing a result of \textit{J. Roitberg} [Topology Appl. 59, No. 3, 261-271 (1994; Zbl 0830.55009)].