Rational homotopy type and self maps

Abstract

\S 1. Introduction.D. Sullivan showed in [1] and [2] that the rational homotopy type of a simply connected simplicial complex can be algebraically described by its minimal model which is constructed from the Q-polynomial forms on it.If the minimal model of a simplicial complex $K$ is isomorphic to the one obtained from its cohomology ring, the rational homotopy type of $K$ is called a formal consequence of the cohomology ring.Complexes, having such a rational homotopy type, enjoy interesting homotopy properties ([2]).The purpose of this paper is to characterize such complexes by the existence of a certain kind of self maps.Since a finite CW-complex has the same homotopy type as a polyhedron we work in the category of simply connected finite CW-complexes.Our main result is THEOREM.Let $K$ be a simply connected finite CW-complex.Then the fol- lowing three conditions on $K$ are equivalent.(1) The rational homotopy type of $K$ is a formal consequence of the cohomology ring.(2) For any integer $r$ , there exists a multiple $s$ of $r$ and a map $f:K\rightarrow K$ such that $f^{*}=s^{*}Id:H^{*}(K;Z)\rightarrow H^{*}(K;Z)$ .(3) There exists a rational number $t(t\neq 0, \pm 1)$ and a map $F:K_{(0)}\rightarrow K_{(0)}$ such that $F^{*}=t^{*}Id:H^{*}(K_{(0)} ; Z)\rightarrow H^{*}(K_{(0)} ; Z)$ , where $K_{(0)}$ denotes the localization of $K$ at zero and the homomorphism $s^{*}Id$ denotes the homomorphism $s^{i}Id$ for each degree $i$ .