RATIONAL HOMOTOPY OF THE SPACE OF SECTIONS OF A NILPOTENT BUNDLE
RATIONAL HOMOTOPY OF THE SPACE OF SECTIONS OF A NILPOTENT BUNDLE
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DOI：
10.1090/s0002-9947-1982-0667163-8
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发表时间：
1982-02
影响因子：
1.3
通讯作者：
A. Haefliger
中科院分区：
文献类型：
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作者：
A. Haefliger
我们证明了沙利文提出的代数构造确实是幂零丛截面空间有理同伦型的模型。 R. Thom 在他的论文 Uhomologie des espaces fonctionnels 中研究了 X 到给定映射 F 同伦的连续映射的空间 F* 的同伦类型。他从F的Postnikov分解出发，一步步构建了函数空间F*。他还指出了如何构造一个描述 F* 有理同伦类型的微分分级代数。后来，Sullivan 给出了一个代数模型，它用代表 X 的 DG 代数和 F 的最小模型来反映这种构造。本文的目的是按照 Thom 的方法，表明 Sullivan 模型确实是在适当限制下的函数空间模型。与 (3) 一样，我们考虑一个稍微更一般的问题，即确定幂零纤维空间 p: Y -» X 与给定截面 s 的截面 Ts 空间的有理同伦类型。在第 1 节中，我们解释了 Thorn 的几何构造。在第 2 节中，我们描述了幂零空间的阿贝尔伽罗瓦覆盖的代数模型。在第 3 节中，我们展示了沙利文模型如何与几何拟合。感谢审稿人对本文的诸多改进。 1. 截面空间的波斯特尼科夫因式分解。 1.1.令 G 为有限生成的 abehan 群，令 X 为路径连通空间，其上同调群 Hk(X\G) 是为每个 k 有限生成的。为了避免拓扑方面的困难（参见（4）），我们可以在单纯集范畴中工作。命题（汤姆（4））。 Eilenberg-Mac Lane 复形 K(G, m) 中 X 的连续映射的空间 K(G, m)x 同伦等价于 Eilenberg-Mac Lane 空间 Ki = K(Hm-'(X; G), i) 的乘积 nr=rA °。更准确地说，令 x G Hm(K(G; m); G) 为 K(G; m) 的基本类。如果 e:K(G,m)X XX^K(G,m)
We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle. In his paper Uhomologie des espaces fonctionnels, R. Thom studied the homotopy type of the space F* of continuous maps of X into F homotopic to a given map/. Starting from a Postnikov decomposition of F, he built the functional space F* step by step. He also indicated how one could construct a differential graded algebra describing the rational homotopy type of F*. Later on, Sullivan gave an algebraic model which mirrors this construction in terms of a DG-algebra representing X and the minimal model of F. The aim of this paper is to show, following the method of Thom, that the model of Sullivan is indeed a model for the functional space under suitable restrictions. As in (3), we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections Ts of a nilpotent fiber space p: Y -» X homotopic to a given section s. In §1 we explain Thorn's geometric construction. In §2 we describe an algebraic model for an abelian Galois covering of a nilpotent space. In §3 we show how the model of Sullivan fits with the geometry. I thank the referee for many improvements of this paper. 1. A Postnikov factorization of the space of sections. 1.1. Let G be a finitely generated abehan group and let X be a path connected space whose cohomology groups Hk(X\ G) are finitely generated for each k. To avoid difficulties with the topologies (cf. (4)), we can work in the category of simplicial sets. Proposition (Thom (4)). The space K(G, m)x of continuous maps of X in the Eilenberg-Mac Lane complex K(G, m) is homotopically equivalent to the product nr=rA °fthe Eilenberg-Mac Lane spaces Ki = K(Hm-'(X; G), i). More precisely, let x G Hm(K(G; m); G) be the fundamental class of K(G; m). If e:K(G,m)X XX^K(G,m)