Unipotency and nilpotency in homotopy equivalences
Unipotency and nilpotency in homotopy equivalences
复制标题
同伦等价中的单能性和零能性
DOI：
10.1016/0040-9383(79)90002-8
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
A. Zabrodsky
中科院分区：
文献类型：
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作者：
E. Dror;A. Zabrodsky
导言这项工作涉及离散群同伦自等价的一个给定的空间X以及完整的拓扑幺半群的某些自我等价采取通常的紧开拓扑。在整个文件中，我们的空间X将是连接的，并且要么是有限维的，要么是有限数量的非平凡同伦群（后者将被称为Postnikov片段或部分）。所有自等价的拓扑幺半群记为aut X，在同伦理论中起着重要的理论作用，因为它的分类空间B aut X将所有与给定纤维X同伦等价的纤维化分类到纤维同伦。关于这个幺半群，我们感兴趣的是以下问题：什么组同伦自等价是幂零的？对于什么样的子幺半群Y g aut X，分类空间BY是幂零空间？设AUT”X为点映射X-+ X的点同伦类的幺半群[X，X10]的离散子群，它由所有的自等价类组成。那么下面的结果可以作为本文的座右铭：
INTRODUCTION THIS WORK concerns the discrete group of homotopy self equivalences of a given space X as well as the full topological monoid of certain self equivalences taken with the usual compact-open topology. Throughout the paper our spaces X will be connected and either finite dimensional or with finite number of non-trivial homotopy groups (the latter will be called Postnikov pieces or sections). The topological monoid of all self equivalences denoted by aut X, plays an important theoretical role in homotopy theory since its classifying space B aut X classifies up to fibre homotopy equivalence all fibrations with a given fibre X. Concerning this monoid we are interested in the following questions: What groups of homotopy self equivalences are nilpotent? For what submonoids Y g aut X is the classifying space BY a nilpotent space? Let us denote by AUT” X the discrete subgroup of the monoid [X, Xl0 of pointed homotopy classes of pointed maps X-+ X, which consist of all the classes of self equivalences. Then the following result can serve as the motto of the paper: