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# Homology, Homotopy and Applications

## Volume 1 (1999)

### Number 1

### On spaces of the same strong $n$-type

Pages: 205 – 217

DOI: http://dx.doi.org/10.4310/HHA.1999.v1.n1.a10

#### Authors

#### Abstract

Let $X$ be a connected CW complex and $[X]$ be its homotopy type. As usual, $\mbox{SNT}(X)$ denotes the pointed set of homotopy types of CW complexes $Y$ such that their $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent for all $n$. In this paper we study a particularly interesting subset of $\mbox{SNT}(X)$, denoted $\mbox{SNT}_{\pi}(X)$, of strong $n$ type; the $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent by homotopy equivalences satisfying an extra condition at the level of homotopy groups. First, we construct a CW complex $X$ such that $\mbox{SNT}_\pi(X) \neq \{ [X] \}$ and we establish a connection between the pointed set $\mbox{SNT}_{\pi}(X)$ and sub-groups of homotopy classes of self-equivalences via a certain $\displaystyle{\lim_{\leftarrow}}^1$ set. Secondly, we prove a conjecture of Arkowitz and Maruyama concerning subgroups of the group of self equivalences of a finite CW complex and we use this result to establish a characterization of simply connected CW complexes with finite dimensional rational cohomology such that $\mbox{SNT}_{\pi}(X) = \{[X]\}$.

#### Keywords

derived functor, Postnikov tower, Self-equivalences, lim$^1$, Phantom map

#### 2010 Mathematics Subject Classification

18G55, 55P10, 55P15