On the same \(n\)-types for the wedges of the Eilenberg-MacLane spaces
We say that two connected \(CW\)-complexes \(X\) and \(X'\) have the same \(n\)-type iff \(X(n) \simeq X'(n)\), for all \(n \in \mathbb N\), where \(X(n)\) is the \(n\)-th Postnikov approximation of \(X\). J. H. C. Whitehead asked: ``If two \(CW\)-complexes are of the same \(n\)-type for all \(n \in \mathbb N\), are they necessarily of the same homotopy type ?'' In [Proc. Camb. Philos. Soc. 53, 922--923 (1957; Zbl 0089.39101)], \textit{J. Adams} showed that the answer to this question is no (see also [\textit{B. I. Gray}, Topology 5, 241--243 (1966; Zbl 0149.20102) and \textit{C. A. McGibbon} and \textit{J. M. Møller}, ibid. 31, No. 1, 177--201 (1992; Zbl 0765.55010)]). Given a connected \(CW\)-complex \(X\), let \(SNT(X)\) be the class of all the homotopy types of \(X'\) with \(X(n) \simeq X'(n)\), for all \(n \in \mathbb N\). It is easily seen that in the case where the dimension of \(X\) is finite or \(X\) has only a finite number of nontrivial homotopy groups, the answer to the Whitehead question is positive. The same holds if all the homotopy groups of \(X\) are finite [Adams, loc. cit.]. Among other results, it is known that \(SNT(\sum^{k} K(\mathbb{Z},2b+1)) = \ast\), for \(k \geq 0\), (where \(\sum^{k} K(\mathbb{Z},2b+1)\) is the \(k\)-th iterated suspension of the Eilenberg-MacLane space \(K(\mathbb{Z},2b+1)\)), cf. [\textit{C. A. McGibbon} and \textit{J. M. Møller}, Lect. Notes Math. 1509, 285--293 (1992; Zbl 0768.55008)]. In this paper the author proves (Theorem 1.1) that if \(Y = K(\mathbb{Z},2a_1) \vee K(\mathbb{Z},2a_2) \vee \ldots \vee K(\mathbb{Z},2a_k)\) is the wedge of Eilenberg-MacLane spaces, where each \(a_i\) (\(i = 1, 2, \ldots , k\)) is a positive integer and \(a_1 < a_2 < \cdots < a_k\), then \(SNT (\sum Y) = \ast\). This result generalizes the previous result of the author that \(SNT(\sum K(\mathbb{Z},2a)) = \ast\) (see [\textit{D.-W. Lee}, Proc. Am. Math. Soc. 137, No. 3, 1161--1168 (2009; Zbl 1162.55006)]).
scientific article; zbMATH DE number 6700372
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