## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 1, Number 1 (1999), 205-217.

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

Yves Fèlix and Jean-Claude Thomas

#### 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 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 establishe 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 $\mbox{SNT}_\pi(X) = \{[X]\}$.

#### Article information

**Source**

Homology Homotopy Appl., Volume 1, Number 1 (1999), 205-217.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139840204

**Mathematical Reviews number (MathSciNet)**

MR1797542

**Zentralblatt MATH identifier**

0952.55005

**Subjects**

Primary: 55P10: Homotopy equivalences

Secondary: 55P15: Classification of homotopy type

#### Citation

Fèlix, Yves; Thomas, Jean-Claude. On spaces of the same strong $n$-type. Homology Homotopy Appl. 1 (1999), no. 1, 205--217. https://projecteuclid.org/euclid.hha/1139840204