Atomicity for Anick's spaces (Q2254794)

A simply connected space \(X\) is atomic if, whenever a self-map \(f: X\to X\) induces an isomorphism in the least non-vanishing degree in homology, it is a homotopy equivalence. Assume that \(p\) is an odd prime and assume that all spaces are localized at \(p\). In the context of the seminal work of Cohen-Moore-Neisendorfer, Anick proved the atomicity of the space \(T^{2n+1}(p^r)\) for all \(p>5\) and all \(r, n\geq 1\). In this paper the author proves the atomicity of the single loop \(\Omega T^{2n+1}(p^r)\) (resp., double loop \(\Omega^2T^{2n+1}(p^r)\)) when \(p\geq 5\) and \(n\geq 2\) or \(p=3\) and \(n+1\neq 3^i\) (resp., when \(p\geq 5\) and \(n\geq 3\) or \(p=3\) and \(n+ 1\neq 3^i\)). In the first section of this paper the author gives a valuable criterion for when a self-map of a simply connected space is a homotopy equivalence.