TITLE: "Localizations of abelian Eilenberg--Mac Lane spaces of finite type"
AUTHORS:
Carles Casacuberta
Universitat Autonoma de Barcelona
08193 Bellaterra, Spain
casac@mat.uab.es
http://mat.uab.es/casac
Jose L. Rodriguez
Universitat Autonoma de Barcelona
08193 Bellaterra, Spain
jlrodri@mat.uab.es
http://mat.uab.es/jlrodri
Jin-Yen Tai
Department of Mathematics, Dartmouth College,
Hanover, NH 03755-3551,
Jin-Yen.Tai@Dartmouth.edu
ABSTRACT:
Using recent techniques of unstable localization, we extend earlier
results on homological localizations of Eilenberg--Mac Lane
spaces, and show that several deep properties of such
localizations can be explained by the preservation of certain
algebraic structures under the effect of idempotent functors.
We study localizations $L_fK(G,n)$ of Eilenberg--Mac Lane
spaces with respect to any map $f$, where $n\ge 1$ and
$G$ is abelian. We find that, if $G$ is finitely generated,
then the result is a $K(A,n)$, where $A$ can be computed using
cohomological data derived from $f$. If $G=\Z$, then $A$ is a
commutative ring which is isomorphic to the ring $\End(A)$
of its own additive endomorphisms; such rings, which we call rigid,
form a proper class which contains the set of solid rings.
From this fact it follows that there is a proper class
of distinct homotopical localizations of the circle $S^1$.
Among other applications of our results, we show that,
if $X$ is a product of abelian Eilenberg--Mac Lane spaces
and $f$ is any map, then the homotopy groups
$\pi_m(L_f X)$ become modules over the ring $\pi_1(L_f S^1)$.