LAUSR

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.