On localization of connective covers (Q1579881)

In the category of pointed simplicial sets \(S.sets*\) the author proves a result of Neisendorfer: ``For every connected \(p\)-complete finite complex \(Y\) with \(\pi_2Y\) torsion, the \(p\)-completion of \(P_{K(Z/p,1)} (Y\langle m\rangle)\) and \(Y\) are of the same homotopy type for any positive integer \(m\)'' without the use of Miller's theorem of Sullivan's conjecture. So, for a given space \(Y\) and any positive integers \(m\neq n\) he considers a map \(f:A\to B\) between connected spaces, \(Y\langle m\rangle\) the \(m\)-connective cover of the space \(Y\), and he gives a general criterion for \(L_f(Y\langle m\rangle)\) and \(L_f (Y\langle n\rangle)\) to have the same homotopy type. Then he applies this criterion to simply connected \(p\)-torsion and simply connected \(p\)-local spaces. Finally, he considers \(f\)-localization in a subcategory of \(S.sets*\) and discusses the result of Neisendorfer.