Cellular approximations using Moore spaces

Rodríguez, J.L.; Scherer, Jérôme

doi:10.1007/978-3-0348-8312-2_22

Cited by 23 publications

(38 citation statements)

References 16 publications

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“…A similar construction is often denoted in topological contexts C W A X since it is inspired by the classical CW-approximation given by Whitehead, see [18,9,7]. Our description relies on [18], and we give only a rough sketch.…”

Section: C Constructing An A-cellular Approximationmentioning

confidence: 97%

“…The notion of approximation appears in [4,8,18] and in other works and is closely related to the notion of a generator as in [11], Proposition V.1. By definition, a generator is an object A (in a given category) that detects equivalences in the sense that if a given map f : X → Y induces, via composition with f , an equivalence hom(A, f ): hom(A, X ) → hom(A, Y ), on the collection of all morphisms, then f is itself an equivalence, where the notion of 'equivalence' must, of course, be specified.…”

Section: A Background Definitions and Examplesmentioning

confidence: 97%

“…As above, given groups X and Y , we denote by Hom(X, Y ) the set of group maps (= homomorphisms) from X to Y and End(X ) = Hom(X, X ). Consider now the notion of an A-cellular object in the category of groups [18]. Here are some examples.…”

Section: A A-equivalence and A-cellular Groupsmentioning

confidence: 98%

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Cellular covers of groups

Farjoun

Göbel

Segev

2007

Journal of Pure and Applied Algebra

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In this paper we discuss the concept of cellular cover for groups, especially nilpotent and finite groups. A cellular cover is a group homomorphism c: G → M such that composition with c induces an isomorphism of sets between Hom(G, G) and Hom(G, M). An interesting example is when G is the universal central extension of the perfect group M. This concept originates in algebraic topology and homological algebra, where it is related to the study of localizations of spaces and other objects. As explained below, it is closely related to the concept of cellular approximation of any group by a given fixed group. We are particularly interested in properties of M that are inherited by G, and in some cases by properties of the kernel of the map c.

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Section: C Constructing An A-cellular Approximationmentioning

confidence: 97%

Section: A Background Definitions and Examplesmentioning

confidence: 97%

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Cellular covers of groups

Farjoun

Göbel

Segev

2007

Journal of Pure and Applied Algebra

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“…The crucial definition and the basic properties of E-rings (for R = Z) come from the seminal paper of Schultz [27] from 1973. Some years later it became clear that E-rings are not only useful in module theory, but also have fruitful applications in homotopy theory and in non-commutative group theory; see [8,9,14,16,23,26]. They are useful tools in the work of Pierce (see [25]), and the existence of arbitrarily large indecomposable E-rings was first shown in [5].…”

Section: Introductionmentioning

confidence: 99%

E(R)-algebras that are sharply transitive modules

Göbel

Herden

2007

Journal of Algebra

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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 pG 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 Z A (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.

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“…A line of research of great relevance in the last years, and very related to our work, is cellular approximation in the category of groups ( [RS01], [Flo07], [DGS07], [DGS08]). …”

Section: Introductionmentioning

confidence: 99%