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 f K(G, n) of Eilenberg-Mac Lane spaces with respect to any map f , where n ≥ 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 π m (L f X) become modules over the ring π 1 (L f S 1 ).
IntroductionWe refer the reader to [5] and [15] for the essentials of homotopical localization.Given any map f : W → V between CW-complexes, for each space X there is a map η: X → L f X where L f X is universal in the homotopy category with the property that the map of function spacesinduced by f is a weak homotopy equivalence. The space L f X is called the f -localization of X.As explained in [15, 4.B], if f is any map, G is any abelian group, and n ≥ 1, then the f -localization of an Eilenberg-Mac Lane space K(G, n) has at most two nontrivial homotopy groups, and it is in fact a product K(A, n) × K(B, n + 1). In the present article, for any map f : W → V , we firstly compute the homotopy groups A and B of the f -localization L f K(G, n) in terms of f when G is finitely generated; in fact, in this case it happens that the group B is necessarily zero.Since localization commutes with finite direct products, it suffices to consider the case when G is cyclic. If G = Z/p r , then we prove that either A = 0 or A = Z/p j for some j ≤ r. This was first observed by Chachólski in the case when V is contractible; under this assumption on V , one finds that necessarily j = r.Contrary to this fact, if no restriction is imposed on the spaces V and W , then each j ≤ r can occur, and the value of j is determined by two (possibly infinite)