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arXiv 1408.4871math.AT
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自同伦等价群上的若干个数

Certain number on the groups of self homotopy equivalences

Ho Won Choi, Kee Young Lee

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9 pages
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摘要：

对于连通基空间$X$，设$[X,X]$是$X$的保基点自映射的所有基同伦类的集合，$\E(X)$是$X$的自同伦等价群.我们用$\A_{\sharp}^k(X)$表示$X$的自映射的同伦类的集合，这些同伦类诱导$i=0,1,\cdots,k$的$π_i(X)$的自同构。也就是说，$[f]\in \A_{\sharp}^k(X)$当且仅当$π_i(f):π_i(X)\toπ_i(X)$是$i=0,1,\cdots ,k$的同构。然后，对于非负整数$k$，求$\E(X)\subseteq\A_{\sharp}^k(X)\subseteq [X,X]$。此外，对于连通CW-复形$X$，我们有$\E(X)=\A_{\sharp}(X)$。本文研究了$\A_{\sharp}^k(X)$的性质，讨论了$\E(X)=\A_{\sharp}^k(X)$的条件和$k$的最小值。此外，我们确定的$k$的各种空间，包括领域，产品的空间，摩尔空间。

英文摘要：

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 $\A_{\sharp}^k(X)$ the set of homotopy classes of self-maps of $X$ that induce an automorphism of $π_i(X)$ for $i=0,1,\cdots,k$. That is, $[f]\in \A_{\sharp}^k(X)$ if and only if $π_i(f):π_i(X)\toπ_i(X)$ is an isomorphism for $i=0,1,\cdots ,k$. Then, $\E(X)\subseteq\A_{\sharp}^k(X)\subseteq [X,X]$ for a nonnegative integer $k$. Moreover, for a connected CW-complex $X$, we have $\E(X)=\A_{\sharp}(X)$. In this paper, we study the properties of $\A_{\sharp}^k(X)$ and discuss the conditions under which $\E(X)=\A_{\sharp}^k(X)$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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