Atomicity for Anick's spaces

It is known that Anick's space T2n+1(pr)T2n+1(pr) is atomic for p≥3p≥3 and r≥1r≥1. We show that ΩT2n+1(pr)ΩT2n+1(pr) and Ω2T2n+1(pr)Ω2T2n+1(pr) are also atomic. The method also applies to show the atomicity of spaces that geometrically realize natural homological filtrations of T2n+1(pr)T2n+1(pr) and ΩT2n+1(pr)ΩT2n+1(pr).