A version of Zabrodsky’s lemma

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)

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

A version of Zabrodsky’s lemma

Cited by 2 publications

References 3 publications

Central extensions and generalized plus-constructions

Central extensions and generalized plus-constructions

Localizations of abelian Eilenberg–Mac Lane spaces of finite type

Contact Info

Product

Resources

About