Rational homotopy type and self maps
Rational homotopy type and self maps
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有理同伦型和自映射
DOI：
10.2969/jmsj/03130427
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发表时间：
1979
影响因子：
0.7
通讯作者：
H. Shiga
中科院分区：
文献类型：
--
作者：
H. Shiga
D. Sullivan在[1]和[2]中证明了单连通简单复合体的有理同伦型可以用其最小模型进行代数描述，该最小模型由其上的q -多项式形式构造而成。如果一个简单复合体$K$的极小模型与其上同构环的极小模型是同构的，则$K$的有理同伦型称为上同构环的形式推论。具有这种有理同伦类型的配合物具有有趣的同伦性质（[2]）。本文的目的是通过某种自映射的存在来表征这种复合体。由于有限cw -复形与多面体具有相同的同伦类型，我们在单连通有限cw -复形的范畴内进行研究。我们的主要结果是定理。设$K$为单连通有限cw复形。那么$K$上的以下三个条件是等价的。(1) $K$的有理同伦型是上同伦环的一个形式推论。(2)对于任意整数$r$，存在$r$的倍数$s$和映射$f:K\rightarrow K$，使得$f^{*}=s^{*}Id:H^{*}(K;Z)\rightarrow H^{*}(K;Z)$。(3)存在有理数$t(t\neq 0, \pm 1)$和地图$F:K_{(0)}\rightarrow K_{(0)}$，使得$F^{*}=t^{*}Id:H^{*}(K_{(0)} ; Z)\rightarrow H^{*}(K_{(0)} ; Z)$，其中$K_{(0)}$表示的定位
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 following 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