On spaces of the same strong $n$-type

Yves Fèlix ; Jean-Claude Thomas

Homology, homotopy, and applications, Tome 1 (1999) no. 1, pp. 205-217.

Résumé

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

Détail

DOI : 10.4310/HHA.1999.v1.n1.a10

Classification : 18G55, 55P10, 55P15
Keywords: derived functor, Postnikov tower, Self-equivalences, lim$^1$, Phantom map