Papers by José Luis Rodríguez Blancas
In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer,... more In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer, but equipped with non necessarily continuous triangle functions. Such spaces endow a generalized topology that is Fréchet-separable, translationinvariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.
Journal of Group Theory, 2017
A group homomorphism e : H → G {e:H\to G} is a cellular cover of G if for every homomorphism φ : ... more A group homomorphism e : H → G {e:H\to G} is a cellular cover of G if for every homomorphism φ : H → G {\varphi:H\to G} there is a unique homomorphism φ ¯ : H → H {\bar{\varphi}:H\to H} such that φ ¯ e = φ {\bar{\varphi}e=\varphi} . Group localizations are defined dually. The main purpose of this paper is to establish 2 ℵ 0 {2^{\aleph_{0}}} varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Göbel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 2 ℵ 0 {2^{\aleph_{0}}} varieties not closed under localizations. Finally, the existence of 2 ℵ 0 {2^{\aleph_{0}}} varieties of groups neither closed under cellular covers nor under localizations is presented as well.
The possibilities of using the new software of virtual reality NeoTrie VR (or Neotrie) to teach g... more The possibilities of using the new software of virtual reality NeoTrie VR (or Neotrie) to teach geometry to children aged 11-14 years old are presented. This software is tested for the first time as a part of math lessons at a School in Żernica (Poland). The use of Neotrie has made it easier to eliminate some of the problems faced by the pupils in the early stages of geometry learning. Moreover, the participating students solve geometric tasks more quickly, compared with previous years, especially those that require spatial imagination and those with a higher degree of difficulty. Since the program has been implemented, pupils are more active and prone to cooperation, and to formulate conclusions. Neotrie turns out to be a useful tool for teaching Geometry, it helps students to better organize their geometric knowledge in a ludic way, to increase their spatial reasoning and creativity.
The motivation of this paper is a suggestion by Höle of comparing the notions of D-boundedness an... more The motivation of this paper is a suggestion by Höle of comparing the notions of D-boundedness and boundedness in Probabilistic Normed spaces (briefly PN spaces), with non necessarily continuous triangle functions. Such spaces are here called “pre-PN spaces”. Some results on Šerstnev spaces due to B. Lafuerza, J. A. Rodŕıguez, and C. Sempi, are here extended to generalized Šerstnev spaces (these are pre-PN spaces satisfying a more general Šerstnev condition). We also prove some facts on PN spaces (with continuous triangle functions). First, a connection between fuzzy normed spaces defined by Felbin and certain Šerstnev PN spaces is established. We further observe that topological vector PN spaces are F -normable and paranormable, and also that locally convex topological vector PN spaces are bornological. This last fact allows to describe continuous linear operators between certain generalized Šerstnev spaces in terms of bounded subsets. 2000 Mathematics Subject Classification: 54E70...
Publicacions Matemàtiques, 2021
We show that in the category of groups, every singly-generated class which is closed under isomor... more We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes. 1 Introduction In a category of R-modules over a certain ring R, the study of the closure of different classes under different operations has been a hot research topic for decades (see [15] for a thorough approach). The operations include extensions, subgroups, quotients, etc. In the case of R = Z, we are dealing, of course, with operations in the category of abelian groups. There is another different but related problem. Given a collection D of objects in a class C, the smallest class that contains D and is closed under certain operations is called the class generated by D under these operations. When this class is equal to C, it is said that C is generated by D under the operations, and the elements of D are called generators of C. In this situation, if D consists of only one object, the class C is said to be singly-generated under the operations. Now the question is clear: given a class C defined
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational... more In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H ∗ -acyclic algebra U and we define A −→ A+ as the U -nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic...
Groups, Geometry, and Dynamics, 2020
A group homomorphism i : H → G is a localization of H, if for every homomorphismn ϕ : H → G there... more A group homomorphism i : H → G is a localization of H, if for every homomorphismn ϕ : H → G there exists a unique endomorphism ψ : G → G such that iψ = ϕ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel. 1 Introduction Simple groups are a good source of many kinds of realization theorems. For example, we know from Droste-Giraudet-Göbel [8] that every group can be expressed as the outer automorphism of some simple group. Using Ol'shanskii [25], Obraztsov [24] proved that every abelian group is the center of some infinite quasi-simple group. In [25] we can find Burnside groups of large prime exponent with many extra properties. In this paper we
ICME-13 Monographs, 2017
We started our workshop showing the largest Sierpinski carpet in the world, built by more than 40... more We started our workshop showing the largest Sierpinski carpet in the world, built by more than 40,000 children from 400 centers in 39 countries. The small pieces were assembled in the Palace of Mediterranean Games in Almería on May 13, 2016, with the help of 1000 people. All the information related to this project, including activities on fractals, can be found at http://topologia.wordpress.com/ sierpinski-carpet-project.
Groups, Geometry, and Dynamics, 2018
We connect work done by Enochs, Rada and Hill in module approximation theory with work undertaken... more We connect work done by Enochs, Rada and Hill in module approximation theory with work undertaken by several group theorists and algebraic topologists in the context of homotopical localization and cellularization of spaces. This allows one to consider envelopes and covers of arbitrary groups. We show some characterizing results for certain classes of groups, and present some open questions.
Algebraic & Geometric Topology, 2016
We prove that every homotopical localization of the circle S 1 is an aspherical space whose funda... more We prove that every homotopical localization of the circle S 1 is an aspherical space whose fundamental group A is abelian and admits a ring structure with unit such that the evaluation map End.A/ ! A at the unit is an isomorphism of rings. Since it is known that there is a proper class of nonisomorphic rings with this property, and we show that all occur in this way, it follows that there is a proper class of distinct homotopical localizations of spaces (in spite of the fact that homological localizations form a set). This answers a question asked by Farjoun in the nineties. More generally, 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 any abelian group, and we show that many properties of G are transferred to the homotopy groups of L f K.G; n/. Among other results, we show that, if X is a product of abelian Eilenberg-Mac Lane spaces and f is any map, then the homotopy groups m .L f X / are modules over the ring 1 .L f S 1 / in a canonical way. This explains and generalizes earlier observations made by other authors in the case of homological localizations.
Epsilon Revista De La Sociedad Andaluza De Educacion Matematica Thales, 2013
Resumen: En este artículo presentamos una experiencia de aula, llevada a cabo con alumnos de 3º d... more Resumen: En este artículo presentamos una experiencia de aula, llevada a cabo con alumnos de 3º de ESO, en la que hemos trabajado las simetrías y los movimientos en el plano a través del Arte Nazarí de la Alhambra de Granada. El estudio de las propiedades geométricas subyacentes en este tipo de decoraciones ha sido abordado mediante la confección de mosaicos con hilos de colores.
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational... more In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H Q *-acyclic algebra U and we define A −→ A + as the U-nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π 0 (A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic and Hochschild homologies over the rational.
The motivation of this paper is a suggestion by H\"ole of comparing the notions of $\D$-boun... more The motivation of this paper is a suggestion by H\"ole of comparing the notions of $\D$-boundedness and boundedness in Probabilistic Normed spaces (briefly PN spaces), with non necessarily continuous triangle functions. Such spaces are here called ``pre-PN spaces''. Some results on \v{S}erstnev spaces due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to generalized \v{S}erstnev spaces
In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer,... more In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer, but equipped with non necessarily continuous triangle functions. Such spaces endow a generalized topology that is Fr\'echet-separable, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.
Topology and its Applications, 2000
For homotopical localization with respect to any continuous map, there are results describing the... more For homotopical localization with respect to any continuous map, there are results describing the relations among the localization functors associated to the maps of a given fibration. Here we treat an analogous question in a group-theoretical context: we study localization functors associated to a short exact sequence of groups. We further specialize to a split short exact sequence of groups. In particular, we describe explicitly the localization functors associated to a semidirect product of finitely generated Abelian groups.
K-Theory, 2004
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational... more In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded operads to the framework of cofibrant operads over an arbitrary ring (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. We construct a universal H Q *-acyclic algebra U and we define A −→ A + as the U-nullification of the algebra A. This map induces an isomorphism in Quillen homology and quotients out the maximal perfect ideal of π 0 (A). As an application, we consider for any associative algebra R the plus-constructions of gl(R) in the categories of homotopy Lie and homotopy Leibniz algebras. This gives rise to two new homology theories for associative algebras, namely homotopy cyclic and homotopy Hochschild homologies. Over the rationals these theories coincide with the classical cyclic and Hochschild homologies.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2002
A group homomorphism η : H → G is called a localization of H if every homomorphism ϕ : H → G can ... more A group homomorphism η : H → G is called a localization of H if every homomorphism ϕ : H → G can be 'extended uniquely' to a homomorphism Φ : G → G in the sense that Φη = ϕ. 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 ≥ 10. Emmanuel 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.
Fundamenta Mathematicae, 2004
As it is well known, torsion abelian groups are not preserved by localization functors. However, ... more As it is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by |T | ℵ0 whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, the relationship between localizations of abelian p-groups and their basic subgroups is completely characterized.
Journal of the London Mathematical Society, 1997
Our object of study is the natural tower which, for any given map f : A → B and each space X, sta... more Our object of study is the natural tower which, for any given map f : A → B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E * , yielding e.g. an analogue of Quillen's plus-construction for each E *. We discuss in detail the case of ordinary homology with coefficients in Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f : A → pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M (Z[1/p], 1).
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Papers by José Luis Rodríguez Blancas