Systems of fixed point sets
Systems of fixed point sets
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定点集系统
DOI：
10.1090/s0002-9947-1983-0690052-0
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发表时间：
1983
影响因子：
1.3
通讯作者：
A. Elmendorf
中科院分区：
文献类型：
--
作者：
A. Elmendorf
设G是紧李群.给出了一种利用不动点集的同伦理论信息构造C-空间的规范方法。该构造是范畴条构造的一个特例。应用包括某些分类空间的简单构造，以及C-EilenbergMac Lane空间和Postnikov塔。0.导论.设G是紧李群，X是G-空间. X的等变同伦理论在其不动点集系统中得到了显著的体现，定义为从某个范畴0G到拓扑空间范畴Top的函子。(Our空间将是紧生成的弱Hausdorff;它们可能配备或可能不配备基点，这取决于上下文。这些函子或系统相对于G-空间具有相当大的技术优势;将大多数同伦理论构造应用于它们是很容易的，而在许多情况下，如何在G-空间中进行是不清楚的。本文的目的是提出一种从任何系统恢复一个G-空间的规范方法，该G-空间保持系统的所有同伦理论结构。这使我们能够给出一些标准拓扑结构的简单等变版本，如Eilenberg-Mac Lane空间和Postnikov塔，并简化其他等变结构。主要定理的陈述。G是一个固定的紧李群。定义.标准轨道范畴，记作0G，是一个具有离散对象空间\0G\=（G/77：77 G的闭子群}的拓扑范畴，并且通过要求自然双射（·）Hoxn0ciG/H，G/K）^{G/K）H是同胚来拓扑化G-映射。我们所说的OC-空间是指从0G到Top的连续逆变函子;这些函子以通常的方式构成拓扑范畴的对象。我们还将考虑GG-环、OC-群等，定义类似。定义.设A1是一个G空间。X的不动点集系统，写作$X，是一个定义如下的GG-空间：$*（G/77）= X "，编辑于1981年10月30日收到，修订版于1982年4月26日。1980年数学学科分类主57S10，55N25。
Let G be a compact Lie group. A canonical method is given for constructing a C-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as C-EilenbergMac Lane spaces and Postnikov towers. 0. Introduction. Let G be a compact Lie group and X a G-space. The equivariant homotopy theory of X is reflected to a remarkable extent in its system of fixed point sets, defined as a functor from a certain category 0G to Top, the category of topological spaces. (Our spaces will be compactly generated weak Hausdorff; they may or may not be equipped with a basepoint, depending on the context.) These functors, or systems, have considerable technical advantages over G-spaces; it is easy to apply most homotopy theoretic constructions to them, whereas in many cases it is unclear how to proceed for G-spaces. It is the purpose of this paper to present a canonical way of recovering from any system a G-space which preserves all the homotopy theoretic structure of the system. This allows us to give easy equivariant versions of some standard topological constructions such as Eilenberg-Mac Lane spaces and Postnikov towers, and to simplify other equivariant constructions.1 1. Statements of the main theorems. Throughout, G is a fixed compact Lie group. Definitions. The category of canonical orbits, written 0G, is a topological category with discrete object space \0G\ = (G/77: 77 a closed subgroup of G} and morphisms the G-maps, topologized by requiring the natural bijection (•) Hoxn0ciG/H,G/K)^{G/K)H to be a homeomorphism. By an Oc-space we shall mean a continuous contravariant functor from 0G to Top; these functors form the objects of a topological category in the usual manner. We will also consider GG-rings, Oc-groups, etc., defined similarly. Definition. Let A1 be a G-space. The fixed point set system of X, written $X, is an GG-space defined as follows: $ *( G/77) = X", Received by the editors October 30, 1981 and, in revised form, April 26, 1982. 1980 Mathematics Subject Classification. Primary 57S10, 55N25.