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1999 On spaces of the same strong $n$-type
Yves Fèlix, Jean-Claude Thomas
Homology Homotopy Appl. 1(1): 205-217 (1999).


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


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Yves Fèlix. Jean-Claude Thomas. "On spaces of the same strong $n$-type." Homology Homotopy Appl. 1 (1) 205 - 217, 1999.


Published: 1999
First available in Project Euclid: 13 February 2006

zbMATH: 0952.55005
MathSciNet: MR1797542

Primary: 55P10
Secondary: 55P15

Rights: Copyright © 1999 International Press of Boston


Vol.1 • No. 1 • 1999
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