Homological localizations preserve 1-connectivity (Q2707504)

The problem considered in this paper is whether or not localizations with respect to generalized homology theories preserve 1-connectivity. For \(E\) a spectrum or the associated generalized homology theory, denote \(L_E\) a localization functor which assigns to every space \(X\) a space \(L_E X\) together with a natural map \(X\to L_EX\) which is terminal in the homotopy category among \(E\)-equivalences with source \(X\). The authors prove that if \(X\) is any 1-connected space, then \(L_E X\) is also 1-connected for every generalized homology theory \(E\). This follows by combining the methods of Bousfield with a result proved by Hopkins and Smith (preprint) according to which a \(K(\mathbb{Z},2)\) is never \(E\)-acyclic if \(E\) is nontrivial. The study is made considering torsion and non-torsion homology theory.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].