Homological localizations preserve 1-connectivity
by Carles Casacuberta and Jerome Scherer
Universitat Autonoma de Barcelona Universite de Lausanne
casac@mat.uab.es jerome.scherer@ima.unil.ch
To appear in Contemporary Mathematics, Proceedings of the 1999
Arolla Conference on Algebraic Topology.
AMS Classification numbers: Primary 55P60, 55N20; Secondary 20K21
Every generalized homology theory $E$ yields a localization
functor $L$ that sends the $E$-equivalences to homotopy
equivalences.
We prove that if $X$ is any $1$-connected space, then $LX$
is also $1$-connected, for every generalized homology theory
$E$. This is deduced from a result by Hopkins and Smith stating
that if $K(\Z,2)$ is $E$-acyclic then $E$ is trivial.