Highly divisible cycles in homological Atiyah-Hirzebruch spectral sequences

Let $F_{\ast }$ be a homology theory of finite type and $\{E_{s,t}^r(X)=\frac{Z_{s,t}^r(X)}{B_{s,t}^r(X)};d_{s,t}^r\}$ denote the Atiyah-Hirzebruch spectral sequence for $F_{\ast }$ of a $CW$-complex $X$. In this paper we study elements of the maximal possible order in the groups $E_{n,0}^2(X)\mathrm{/}{Z}_{n,0}^r(X)$.