The same $n$-type structure of the suspension of the wedge products of the Eilenberg-MacLane spaces

TL;DR

This paper proves that the suspended wedge sum of various Eilenberg-MacLane spaces has a unique homotopy type determined by its Postnikov approximations, answering a generalized question about $n$-types.

Contribution

It establishes that the set of all the same $n$-types for the suspension of wedge products of Eilenberg-MacLane spaces contains only one element, extending previous results.

Findings

Unique $n$-type for the suspension of wedge products of Eilenberg-MacLane spaces

Answer to a generalized version of McGibbon and Møller's question

Homotopy type determined by Postnikov approximations

Abstract

For a connected CW-complex, we let $SNT(X)$ be the set of all homotopy types $[Y]$ such that the Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ of $X$ and $Y$, respectively, are homotopy equivalent for all positive integers $n$. In 1992, McGibbon and M{\o}ller (\cite[page 287]{MM}) raised the following question: Is $SNT(\Sigma \mathbb C P^\infty) = *$ or not? In this article, we give an answer to the more generalized version of this query: The set of all the same $n$-types of the suspended wedge sum of the Eilenberg-MacLane spaces of various types of both even and odd integers is the set which consists of only one element as a single homotopy type of itself.