Length of a chain composed by certain monoids of self maps

초록

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 πk(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. © 2020 Elsevier B.V.

키워드