Certain number on the groups of self homotopy equivalences
Certain number on the groups of self homotopy equivalences
复制标题
自同伦等价群上的一定数
DOI：
10.1016/j.topol.2014.12.004
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Keean Lee
中科院分区：
文献类型：
--
作者：
Ho;Keean Lee
对于连通基空间 X，令 [X, X] 为 X 的基点保留自映射的所有基同伦类的集合，并令 E (X) 为 X 的自同伦等价群。我们用 A♯ k (X) 表示 X 的自映射的同伦类集合，其在 i= 0, 1,⋯, k 时诱导 π i (X) 自同构。也就是说，[f]∈ A♯ k (X) 当且仅当 π i (f): π i (X)→ π i (X) 是 i= 0, 1,⋯, k 的同构。然后，E (X)⊆ A♯ k (X)⊆[X, X] 为非负整数 k。此外，对于连通的 CW 复形 X，我们有 E (X)= A♯(X)。本文研究了A♯ k(X)的性质，并讨论了E(X)=A♯ k(X)的条件以及该k的最小值。此外，我们还确定了各种空间的 k 值，包括球体、空间乘积和摩尔空间。
For a connected based space X, let [X, X] be the set of all based homotopy classes of base point preserving self map of X and let E (X) be the group of self-homotopy equivalences of X. We denote by A♯ k (X) the set of homotopy classes of self-maps of X that induce an automorphism of π i (X) for i= 0, 1,⋯, k. That is,[f]∈ A♯ k (X) if and only if π i (f): π i (X)→ π i (X) is an isomorphism for i= 0, 1,⋯, k. Then, E (X)⊆ A♯ k (X)⊆[X, X] for a nonnegative integer k. Moreover, for a connected CW-complex X, we have E (X)= A♯(X). In this paper, we study the properties of A♯ k (X) and discuss the conditions under which E (X)= A♯ k (X) and the minimum value of such k. Furthermore, we determine the value of k for various spaces, including spheres, products of spaces, and Moore spaces.