Abstract For a based CW-complex X, A ♯ n ( X ) is the submonoid of [ X , X ] which consists of all homotopy classes of self-maps of X that induce an automorphism on π k ( X ) for all 0 ≤ k ≤ n . Since, for m n , A ♯ n ( X ) ⊆ A ♯ m ( X ) , there is a chain by inclusions: E ( X ) ⊆ A ♯ ∞ ( X ) ⊆ . . . ⊆ A ♯ 1 ( X ) ⊆ A ♯ 0 ( X ) = [ X , X ] . In this paper, we study the number of strict inclusions in this chain for a given connected CW-complex. We call this number the self-length of a given space. We prove that the self-length is a homotopy invariant and investigate the close connection with the self-closeness number, which is the minimum number n such that E ( X ) = A ♯ n ( X ) . Moreover, we determine self-lengths of several spaces and provide the lower bounds or upper bounds of the self-lengths of some spaces.

中文翻译：

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由自映射的某些幺半群组成的链的长度

摘要 对于基于 CW 的复形 X，A ♯ n ( X ) 是 [ X , X ] 的子幺半群，它由 X 的自映射的所有同伦类组成，这些自映射在 π k ( X ) 上对所有 0 ≤ k ≤ n 。因为，对于 mn ，A ♯ n ( X ) ⊆ A ♯ m ( X ) ，有一个包含链： E ( X ) ⊆ A ♯ ∞ ( X ) ⊆ 。. . ⊆ A ♯ 1 ( X ) ⊆ A ♯ 0 ( X ) = [ X , X ] 。在本文中，我们研究了给定连接的 CW 复合体的该链中严格包含的数量。我们称这个数字为给定空间的自长。我们证明了自长是一个同伦不变量，并研究了与自亲数的紧密联系，自亲数是满足 E ( X ) = A ♯ n ( X ) 的最小数 n。此外，我们确定了几个空间的自长，并提供了一些空间的自长的下限或上限。