Certain number on the groups of self homotopy equivalences

Abstract: 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 $\pi_i(X)$ for $i=0,1,\cdots,k$. That is, $[f]\in \A_{\sharp}^k(X)$ if and only if $\pi_i(f):\pi_i(X)\to\pi_i(X)$ 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.