Dror Farjoun Research Articles

Cellular Covers of Local Groups

We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.

Localization genus

Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.

Fibre sequences and localization of simplicial sheaves

In this paper, we discuss the theory of quasi-fibrations in proper Bousfield localizations of model categories of simplicial sheaves. We provide a construction of fibrewise localization and use this construction to generalize a criterion for locality of fibre sequences due to Berrick and Dror Farjoun. The result allows a better understanding of unstable A^1-homotopy theory.

The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

Let R be any principal ideal domain of cardinality < 2ℵ0 , but not a field. We will construct arbitrarily large E(R)-algebras A (of size ≥ 2ℵ0 ) which are at the same time principal ideal domains over R. It follows that the automorphism group of the R-module R A acts sharply transitive on the pure elements of R A. In answering a question of Emmanuel Dror Farjoun, the existence of such large uniquely transitive (UT-modules) for R = ℤ was shown in Göbel and Shelah (2004). The new method, passing first through ring theory, simplifies the arguments; this idea, using localizations, comes from Herden (2005). We applied it recently in Göbel and Herden (2007b) to find such E(R)-algebras of size ≤ 2ℵ0 .

Formula omitted]-algebras that are sharply transitive modules

We will consider modules over principal ideal domains that are neither fields nor complete discrete valuation rings and show their wild behavior in two instances, which we explain for simplicity in the category of abelian groups. In the first case we consider constructions of large E-rings: This question was answered in 1987 by Dugas, Mader, Vinsonhaler [M. Dugas, A. Mader, C. Vinsonhaler, Large E-rings exist, J. Algebra 108 (1) (1987) 88–101]. Recall that a ring A is an E-ring if the canonical map End Z A → A ( φ ↦ φ ( 1 ) ) is an isomorphism. The second question was raised in 1997 by Emmanuel Dror Farjoun and answered recently in [R. Göbel, S. Shelah, Uniquely transitive torsion-free abelian groups, in: Rings, Modules, Algebras, and Abelian Groups, in: Pure Appl. Math., vol. 236, Marcel Dekker, 2004, pp. 271–290]. There is a (non-trivial) torsion-free abelian group G (other than Z and with pure elements) such that its automorphism group Aut G acts sharply transitive on the pure elements of G. These abelian groups G are called UT-groups (or UT-modules for more general rings). Recall that pure elements of a torsion-free abelian group are those divisible only by ±1 and note that automorphisms leave the family p G of pure elements of G invariant. The two apparently distinct questions, can be unified and strengthened. The link between them is the notion of a pure-invertible ring A, which simply means that all pure elements of A Z (seen as abelian group) are units of the commutative ring A. For technical reasons (to have non-trivial pure elements) we also require that A is ℵ 1 -free as abelian group. In the first part of this paper we transform Farjoun's problem into a ring theoretic question on A. It turns out that A must be a pure-invertible ring and an E-ring at the same time. In the second part of the paper we construct such rings assuming ZFC + CH (the special continuum hypothesis). Thus the existence of UT-groups follows and as a byproduct we obtain a new class of E-rings which are at the same time principal ideal domains.

HOMOTOPY FIXED POINT SETS AND ACTIONS ON HOMOGENEOUS SPACES OF p-COMPACT GROUPS

We generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups.

CONSTRUCTING SIMPLE GROUPS FOR LOCALIZATIONS

ABSTRACT A group homomorphism is called a localization> of if every homomorphism can be extended uniquely' to a homomorphism in the sense that . This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction. For localizations of (almost) commutative structures often resembles properties of , e.g. size or satisfying certain systems of equalities and non-equalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if is a finite (non-abelian) group. Libman showed that for a natural embedding of the alternating group is a localization if is even and . Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of 's. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space for many groups . The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].

Large localizations of finite simple groups

A group homomorphism eta:H-->G is called a localization of H if every homomorphism phi:H-->G can be `extended uniquely' to a homomorphism Phi:G-->G in the sense that Phi eta=phi. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation A_n-->SO_{n-1}(R) of the alternating group A_n, which turns out to be a localization for n even and n>9. Dror Farjoun asked if there is any upper bound in cardinality for localizations of A_n. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group H, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space K(H,1) for any non abelian finite simple group H.

Homotopical localizations of spaces

For a map f of spaces, Dror Farjoun and the author have constructed an f -localization functor, where a space Y is called f -local when map( f, Y ) is an equivalence. This very general construction gives all known idempotent homotopy functors of spaces. The main theorem of this paper shows that f -localization functors always preserve fiber sequences of connected H-spaces up to small error terms. For instance, the localization with respect to the n th Morava K -theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups. This implies, for example, that a K (1)-homology equivalence of H -spaces must induce an isomorphism of v 1 -periodic homotopy groups. Results are also obtained on the A -nullification or A -periodization functors, which are just the f -localization functors for the maps f from spaces A to points. Two spaces are said to have the same nullity when they give the same nullification functors, and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds. The A -nullifications of nilpotent Postnikov spaces are completely determined.

Localization and periodicity in unstable homotopy theory

In this paper, we develop a hierarchy of natural localizations of spaces, called the vn-periodizations for n > 0, which may be used to expose and study periodic phenomena in unstable homotopy theory. These vn-periodizations act less radically than the corresponding homological localizations [2] and respect fibrations to a very considerable extent. A major part of this paper is devoted to developing the general theory of periodizations of spaces, thereby providing a foundation for the study of the vn-periodization and many others. Some of this general theory has been developed independently by Dror Farjoun [13], [14], and we refer the reader to his work for an alternative approach with other interesting general results. During the past decade, remarkable progress was made by Ravenel, Hopkins, Devinatz, and Smith [12], [16], [17], [33] toward a global understanding of stable periodic phenomena, and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena. An excellent exposition of localization and periodicity in stable homotopy theory is now available in Ravenel's book [35]. Major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena. These phenomena belong to a hierarchy starting with those detected rationally, followed by those detected in classical K-theory and in the successive Morava K-theories. Most fundamentally, each finite CW-spectrum has an intrinsic periodicity given by a vn self-map which becomes a selfequivalence after suitable localization. For simplicity, we describe our results in the pointed homotopy category Hoo of connected CW-complexes. For spaces W, Y E Hoo , we call Y W-periodic or W-local when the pointed mapping space map. (W, Y) is contractible, or equivalently when [Xt W, Y] * for t > 0. As shown more generally in [3, Corollary 7.2], [10], or [13], there is a natural initial example X -+ PwX of a map from X to a W-periodic space PwX in Hoo, and we call this the W-periodization or W-localization of X. Our notation is derived from the classical example where W = Sn+l and PwX is the n-th Postnikov section