The Dror-Whitehead theorem in prohomotopy and shape theories
The Dror-Whitehead theorem in prohomotopy and shape theories
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原同伦和形状理论中的 Dror-Whitehead 定理
DOI：
10.1090/s0002-9947-1981-0632540-7
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
S. Singh
中科院分区：
文献类型：
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作者：
S. Singh
许多类似的经典怀特海定理从同伦理论现在可以在亲同伦和形状理论。E. Dror将Whitehead定理的同调版本从著名的单连通情形扩展到更一般的情形，例如幂零情形。我们证明了一个完整的模拟Dror的定理在亲同伦和形状理论。更具体地说，假设f：X -+ Y是点连通拓扑空间的原同伦范畴中的一个态射，它诱导整同调原群的同构。然后f诱导同伦亲群的同构，例如，当X和Y是单的、幂零的、完备的或H-对象时;这些概念在同伦理论中是众所周知的，我们自然地将它们扩展到亲同伦和形状理论。0.导论.本文的目的是建立一个类似的Dror的推广的Whitehead定理（见[DR]），下面陈述为DrorWhitehead定理，在上下文中的亲同伦和形状理论。我们将在下面几段详细阐述这些问题。德罗尔-怀特海定理设映射f：X -* Y诱导空间X和Y的所有整系数同调群的同构。则f诱导所有同伦群的同构当且仅当F. 97 Tf是满态，1 ′ j，Tf是单态，rTf是单态。函子r1 '，r，和P是通过考虑同伦群上的基本群的作用来定义的;见[DR]或？2这张纸一类重要的空间，其中博士，怀特海定理适用是幂零空间;见[DR]，[（]，[BK]或？3这张纸本文的定理（4.1.3）是我们将Dror-Whitehead定理推广到原同伦和形理论的结果;也参见定理（4.1.1）、（4.1.2）、（4.1.3）、（4.5.1）和推论（4.3）。我们的推论（4.3）以与Dror推广Whitehead定理相同的方式推广了Raussen [RA]的一个定理。一个平行发展的关键成分的“亲代数”是由[SIJ]提供的扩展工作的Stallings [ST]和Dror [DR]关于代数;和我们的整个计划是一个自然延伸Dror的工作。作为结束语，我们可以补充说，经典怀特黑德定理的各种版本的许多类似物已经在1980年9月3日和1980年12月8日编辑收到的pro-homotopy中进行了研究。1980年数学学科分类。初级55 P55、55 Q 07;次级54 C56、55 N 05。
Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror's theorems in pro-homotopy and shape theories. More specifically, suppose f: X -+ Y is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then f induces isomorphisms of the homotopy pro-groups, for instance, when X and Y are simple, nilpotent, complete, or H-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories. 0. Introduction. The purpose of this paper is to establish an analogue of Dror's generalization of the Whitehead theorem (see [DR]), stated below as the DrorWhitehead theorem, in the context of the pro-homotopy and shape theories. We shall elaborate on these matters in the next few paragraphs. THE DROR-WHITEHEAD THEOREM. Suppose a map f: X -* Y induces isomorphisms of all the homology groups of spaces X and Y with integral coefficients. Then f induces isomorphisms of all the homotopy groups if and only if F.97Tf is an epimorphism, 1'j,Tf is a monomorphism, and rTf is a monomorphism. The functors r1',, r,, and P are defined by considering the action of the fundamental group on the homotopy groups; see [DR] or ?2 of this paper. An important class of spaces to which the Dror-Whitehead theorem applies is nilpotent spaces; see [DR], [(], [BK] or ?3 of this paper. Theorem (4.1.3) of this paper is our extension of the Dror-Whitehead theorem to pro-homotopy and shape theories; also, see Theorems (4.1.1), (4.1.2), (4.1.3), (4.5.1), and Corollary (4.3). Our Corollary (4.3) extends a theorem of Raussen [RA] in the same manner as Dror extends the Whitehead theorem. A parallel development of pivotal ingredients of "pro-algebra" is provided by [SIJ] which extends the work of Stallings [ST] and Dror [DR] concerning algebra; and our entire program is a natural extension of Dror's work. As a concluding remark, we may add that many analogues of the various versions of the classical Whitehead theorem have been studied in pro-homotopy Received by the editors September 3, 1980 and, in revised form, December 8, 1980. 1980 Mathemnatics Subject Classification. Primary 55P55, 55Q07; Secondary 54C56, 55N05.