Contents Online
Homology, Homotopy and Applications
Volume 1 (1999)
Number 1
On spaces of the same strong $n$-type
Pages: 205 – 217
DOI: https://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
Published 1 January 1999
An erratum to this article is available as HHA 2(1) pp. 115-118.