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The aim of this note is to shed some light on the topological structure of the unstable set of an isolated invariant set K. We give a bound on the number of essential quasicomponents of the unstable set of K in terms of the homological Conley index of K. The proof relies on an explicit pairing between Čech homology classes and Alexander–Spanier cohomology classes that takes the form of an integral., Ministerio de Ciencia, Innovación y Universidades, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let WCR2 be an open subset and f :W-f ðWÞCR2 be an orientation reversing homeomorphism. We prove that if pAW is an isolated and stable fixed point of f then the fixed point index of f at p; iR2 ð f ; pÞ; is 1. We apply our theorem to the study of the orbital stability of isolated periodic orbits of flows in four-dimensional riemannian manifolds., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

We introduce the notion of internal fundamental sequence and prove that any shape morphism from an arbitrary compactum X to an internally movable compactum Y is induced by an internal fundamental sequence. We use this special kind of fundamental sequences to give characterizations and some properties of AANRc-sets and AANR,-sets. The paper ends with a section devoted to internal FANR’s., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism phi:A --> R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or C(m)-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal)., DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

For an abelian topological group G, let G∧ denote the character group of G. The group G is called reflexive if the evaluation map is a topological isomorphism of G onto G∧∧, and G is called strongly reflexive if all closed subgroups and quotient groups of G and G∧ are reflexive. In this paper the authors study the relationship of reflexivity (and strong reflexivity) among G, A, and G/K, where A is an open subgroup and K a compact subgroup of G. Strong reflexivity is closely connected with the notion of strong duality introduced by R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32]. In fact, G is strongly reflexive if and only if the natural homomorphism G∧×G→T is a strong duality. R. Venkataraman [Math. Z. 143 (1975), no. 2, 105–112] originally claimed that if G is reflexive, then so is A. However, his proof includes inaccuracies. The present paper includes a new proof in this regard. In all, the following theorems are proved. Theorem 1: G is reflexive [resp. strongly reflexive] if and only if A is reflexive [resp. strongly reflexive]. Theorem 2: If G admits sufficiently many continuous characters and G/K is reflexive [resp. strongly reflexive], then G is reflexive [resp. strongly reflexive]. Conversely, if G is reflexive and K is dually closed in G, then G/K is reflexive. Theorem 3: Every closed subgroup H and the quotient group G/H of a strongly reflexive group G are strongly reflexive, D.G.I.C.Y.T., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The main goal of this paper is to prove that for each n > 2, every sequence of integers satisfying Dold’s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving Rn-homeomorphism at an isolated stable fixed point. We use Conley index techniques even though stable fixed points are not isolated invariant sets., MICINN, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The author proves that if f is an orientation reversing homeomorphism of the plane and p is an isolated and stable fixed point of f then the fixed point index of f at p is equal to 1 . In the orientation preserving case this result was obtained by E. N. Dancer and R. Ortega [J. Dynam. Differential Equations 6 (1994), no. 4, 631–637. The proof is based on the prime ends compactifications method and a fixed point result by K. M. Kuperberg [Proc. Amer. Math. Soc. 112 (1991), no. 1, 223–229., MCyT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Part of this paper is contained in the author's Doctoral Thesis written under the supervisionofProfessor E.Outerelo. Thispaper has been publishedinashorted version in Quest.& Ans. Gen. Topology 5 (1987), 293-302., In this paper -paracompact and well-situated subsets are further examined. A subset E of a space X is paracompact if every covering of E by open sets has a refinement by open sets, locally finite in X, which covers E [C. E. Aull, Proc. 2nd Prague Topol. Symp. 1966, 45-51 (1967; Zbl 0162.264)] and is well-situated in X if for every paracompact T2 space Y, E × Y is -paracompact in X × Y [H. W. Martin, Topology Appl.12, 305-313 (1981; Zbl 0483.54011)]. Covering properties of -paracompact and wellsituated ubsets are obtained, -paracompact and well-situated subsets are characterizedin regular spaces, the behavior of paracompact and well-situated subsets under perfect mappings is studied, and it is shown that the class of all paracompact T2 spaceswhich are well-situated in every paracompact T2 space in which they are embedded as closed subsets, is perfect., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The introduction of intuitionistic fuzzy sets is due to K.T. Atanassov who, also, defined various topological operators for these sets. D. Çoker defined the intuitionistic fuzzy topological spaces. In this paper we show relations between Atanassov’s topological operators and Çoker's intuitionistic fuzzy topology., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A theorem of Glicksberg states that, for an abelian group G, two locally compact topologies with the same set of continuous characters must coincide. In [12] it is asserted that this fact also holds for two Pontryagin reflexive topologies. We prove here that this statement is not correct, and we give some additional conditions under which it is true. We provide some examples of classes of groups determined by their continuous characters., Ministerio de Educación, Xunta de Galicia, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A link in S3 is called a universal link if every closed orientable 3-manifold is a branched cover of S3 over this link. It is well known that the Borromean rings and many other links are universal links. The question whether a link is universal can be naturally extended to orbifolds. An orbifold M is said to be universal if every closed orientable 3-manifold is the underlying space of an orbifold which is an orbifold covering of M. Let Bm,n,p denote the orbifold whose underlying space is S3, whose singular set is the Borromean rings B, and whose isotropy groups for the three components of B are cyclic groups of orders m, n and p. In an earlier paper of H. M. Hilden et al. [Invent. Math. 87 (1987), no. 3, 441–456;], it was shown that B4,4,4 is universal. In this paper, the authors generalize this result and prove that Bm,2p,2q is universal for every m≥3, p≥2, q≥2., MTM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Throughout his paper, the author uses "orientable manifold'' to mean a compact connected orientable 3-manifold without boundary. Such a manifold is known to be a ramified covering over a link of the 3-sphere, in which the ramification index of each singular point is ≤2. If the covering has n leaves, suppose that there are m points of index 2 and 2m points of index 1; such a covering is of type (m,n−2m). The author's main theorem states: Every orientable manifold is a ramified covering of type (1,n−2). He also uses the notion of a "link with a colouring of type (m,n−2m)''; these are intimately related to ramified coverings of type (m,n−2m). He conjectures that every link having a colouring of type (1,n−2) is "separable'', a term too complicated to define here. With this conjecture and his main theorem, he enunciates two further theorems and a second conjecture to show that his two conjectures, if true, would imply the Poincaré hypothesis for 3-manifolds. The author adds a note in proof to say that his first conjecture is false, as will be shown in a forthcoming paper by R. H. Fox. It therefore seems unnecessary to detail the conjectures in this review., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We extend the ideas of Friedman and Qin [5] to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with pg = 0, q > 0 and anticanonical divisor −K effective, for any wall ç with lç = 1 4 (ç2 −p1) being 0 or 1., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

W. Thurston proved the existence of universal links L⊂S3 which are defined by the property that every closed orientable 3-manifold is a branched covering over L⊂S3. The authors answered earlier [Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 449–450;] Thurston's question of whether there are universal knots in the affirmative. In the paper under review, they start from the fact that every closed orientable 3-manifold is an irregular 3-fold covering over a negative closed braid, and proceed by changing the braid by certain moves which do not alter the covering manifold. Thus they arrive at the conclusion that the Whitehead link, the Borromean rings and the knot 946 are universal. Whether the figure-eight knot is universal remains an open question., Comisión Asesora de Investigación científica y técnica, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let hX, di be a metric space. We characterise the family of subsets of X on which each locally Lipschitz function defined on X is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let G be the fundamental group of the complement of the torus knot of type (m, n). This has a presentation G = < x, y vertical bar x(m) = y(n)>. We find the geometric description of the character variety X(G) of characters of representations of G into SL(2, C)., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We study the homotopical and homological properties of the attractors evolving from a generalized Hopf bifurcation. We consider the Lorenz equations for parameter values near the Hopf bifurcation and study a natural Morse decomposition of the global attractor, calculating the Cech homotopy type of the Lorenz attractor, the shape indexes of the Morse sets and the Morse equation of the decomposition., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We prove a gluing formula for the Donaldson invariants of the connected sum of two fourmanifolds along a surface of genus 2. We also prove a finite type condition for manifolds containing a surface of genus 2, self-intersection zero and representing an odd homology class., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Conference on General Topology in Honour of Peter Collins and Mike Reed, AUG 07-10, 2006, Oxford, ENGLAND, We study in this paper preservation of dynamical and shape theoretical properties under Continuation for parametrized families of flows. We show that, although attractors continue. the same does not hold for non-saddle sets. However, when they continue, their shape is preserved in quite general settings, which include differentiable families of flows and regular non-saddle sets for general flows, not necessarily differentiable. We also Study how the continuation of a non-saddle set influences that of its dual non-saddle set., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we study the group of isometries over the order polytope of a poset. We provide a result that characterizes any isometry based on the order structure in the original poset. From this result we provide upper bounds for the number of isometries over the order polytope in terms of its number of connected components. Finally, as an example of application, we recover the set of isometries for the polytope of fuzzy measures and the polytope of p-symmetric measures when the indifference partition is fixed., MEC, FEDER, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

This paper deals with the equivalence between u-density and m-density for the subrings of C(X). It was proved by Kurzweil that such equivalence holds for those subrings that are closed under bounded inversion. Here an example is given in C(N) of a u-dense subring that is not m-dense. It is deduced that the two types of density coincide only in the trivial case where these topologies are the same, that is, if and only if X is a pseudocompact space., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

For a linear sublattice F of C( X), the set of all real continuous functions on the completely regular space X, we denote by A( F) the smallest uniformly closed and inverse-closed subalgebra of C( X) that contains F. In this paper we study different methods to generate A( F) from F. For that, we introduce some families of functions which are defined in terms of suprema or sums of certain countably many functions in F. And we prove that A( F) is the uniform closure of each of these families. We obtain, in particular, a generalization of a known result about the generation of A( F) when F is a uniformly closed linear sublattice of bounded functions., DGES (Spain), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We give the shape of the Donaldson invariants of a general 4-manifold with b1 = 0 and b+ > 1. The resulting expression involves modular forms and matches the physical predictions., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We answer in the affirmative the question posed by Babenko, and Taimanov [3] on the existence of nonformal, simply connected, compact symplectic manifolds of dimension 8., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We describe a new way of obtaining the Morse equations of a Morse decomposition of an isolated invariant set. This is achieved through a filtration of truncated unstable manifolds associated with the decomposition. The results in the paper make it possible to calculate the Morse equations (and also the Conley index) in many interesting situations without using index pairs. We also study the intrinsic topology of the unstable manifold and obtain new duality properties of the cohomological Conley index., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Purpose - In 2005, Smarandache generalized the Atanassov's intuitionistic fuzzy sets (IFSs) to neutrosophic sets (NS), and other researchers introduced the notion of interval neutrosophic set (INSs), which is an instance of NS, and studied various properties. The notion of neutrosophic topology on the non-standard interval is also due to Smarandache. The purpose of this paper is to study relations between INSs and topology. Design/methodology/approach - The paper investigates the possible relations between INSs and topology. Findings - Relations on INSs and neutrosophic topology. Research limitations/implications - Clearly, the paper is confined to IFSs and NSs. Practical implications - The main applications are in the mathematical field. Originality/value - The paper shows original results on fuzzy sets and topology., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Special Issue in Honor of the 100th Anniversary of S.M. Ulam, A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy. A measured solenoid immersed in a smooth manifold produces a closed current (known as a generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (up to scalars) transversal measure. It is known that for any smooth manifold, any real homology class is represented by a uniquely ergodic solenoid. In this paper, we prove that the currents associated to uniquely ergodic solenoids are dense in the space of closed currents,therefore proving the abundance of such objects., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A butterfly is a 3-ball B with an even number of polygonal faces, named wings, pair-wise identified. Each identification between two wings is required to be a topological reflexion whose axis is an edge shared by the wings. The set of axes of the identifications is called the thorax of the butterfly. A knot K⊂S3 admits a butterfly representation if there is a butterfly B with thorax T such that, after the identifications, (B,T) is homeomorphic to (S3,K). In this paper it is shown that any 3-colorable knot admits a butterfly representation (B,T) such that the butterfly B has a 4-colored triangulation compatible with the 3-coloration of the knot. By a result of H. M. Hilden [Amer. J. Math. 98 (1976), no. 4, 989–997;] and J. M. Montesinos [Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;], one can associate to any 3-manifold a 3-colored knot. A corollary of the main result of the paper is therefore that one can associate to any 3-manifold at least one butterfly., COLCIENCIAS, DIME, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We show some examples of compact symplectic solvmanifolds, of dimension greater than four, which are cohomologically Kähler and do not admit Kähler metric since their fundamental groups cannot be the fundamental group of any compact Kähler manifold. Some of the examples that we study were considered by Benson and Gordon (1990). However, whether such manifolds have Kähler metrics was an open question. The formality and the hard Lefschetz property are studied for the symplectic submanifolds constructed by Auroux (1997) and some consequences are discussed., MCYT, Research Training, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We present sufficient conditions on an approximate mapping F : H --> Y of approximate inverse systems in order that the limit f : X --> Y of F is a universal map in the sense of Holsztynski. A similar theorem holds for a more restrictive concept of a proximately universal map introduced recently by the second author. We get as corollaries some sufficient conditions on an approximate inverse system implying that the its limit has the (proximate) fixed point property. In particular, every chainable compact Hausdorff space has the proximate fixed point property., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category HN whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicated with respect and friendship to Professor Yukio Matsumoto on his 60th birthday, In this paper, the possibility of embedding a nontrivial string (R3,K) in the trivial knot (S3,U) is investigated. Uncountably many examples are given. The complementary space in S3 of the image of R3 under the embedding is a continuum. Some well-known snake-like continua appear as these residual spaces. The 2-fold coverings of R3 branched over the strings involved are studied. As a consequence, concrete descriptions of the p-adic solenoids are given, and it is shown that the Whitehead continuum is homeomorphic to Bing's snake-like continuum without end points., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

International Conference on Topology and Its Applications, SEP 02-09, 2000, Ohrid, MACEDONIA, The notion of a non-saddle decomposition of a compact ANR is introduced. This notion extends that of a an attractor-repeller pair. Some cohomological properties of non-saddle decompositions are studied. In particular, some inequalities in the spirit of the Morse-Smale equations for attractor-repeller pairs are obtained. These inequalities involve the ranks of the cohomological Conley index and also of a new cohomological invariant introduced here. The notion of a cyclic Morse decomposition is also introduced and it is proved that this kind of decomposition admits filtrations by non-saddle sets. Finally, these filtrations are used to obtain Morse-Smale equations that generalize those of a Morse decomposition., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The authors prove that every geometric orbifold is good. More precisely, let X be a smooth connected manifold, and let G be a group of diffeomorphisms of X with the property that if any two elements of G agree on a nonempty open subset of X, then they coincide on X. If Q is an orbifold which is locally modelled on quotients of open subsets of X by finite subgroups of G, then the authors prove that the universal orbifold covering of Q is a (G,X)-manifold. A similar theorem was stated, and the proof sketched, in W. Thurston's lecture notes on the geometry and topology of 3-manifolds., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let f : X Y be a map between two spaces for which Y is stratified such that f is a fibration over each open stratum. We find some spectral sequences to compute the homology of X in terms of the homology of Y and that of the fibers. We apply this to give a Lefschetz theorem for the degeneracy loci of a morphism between holomorphic bundles on a complex manifold., Research Training Network, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Forma parte de la "Décima edición del Programa de Promoción de la Cultura Científica y Tecnológica, organizado por la Real Academia de Ciencias Exactas, Físicas y Naturales", La cristalografía es una disciplina de la geología que estudia los cristales: sus formas externas e internas; las posibilidades de estas formas, etc. Los aspectos matemáticos de la teoría son de extraordinario interés para los matemáticos porque admiten generalizaciones en varias direcciones. En primer lugar, pueden estudiarse los grupos cristalográficos n dimensionales para todo n>0. Algunos grupos de dimensiones altas (5 ó más) han aclarado la estructura de algunos cuasicristales. Pero la generalización más interesante consiste en el estudio de los grupos cristalográficos en espacios no euclidianos. El estudio de ellos, en el plano hiperbólico, por ejemplo, tiene relaciones muy importantes (que se remontan a Gauss) con la teoría de números. El caso tridimensional está relacionado con la teoría de nudos, etc. En esta conferencia introduciré estas ideas empleando útiles topológicos e ilustrándolos mediante fotografías de flores, insectos y obras de arte., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Some extension properties of maps defined on dense subsets are studied for approximate polyhedra. The latter are characterised as approximate extensors for finite maps with small oscillation., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In an earlier paper, we proved that Smarandache’s definition of neutrosophic paraconsistent topology is neither a generalization of Çoker’s intuitionistic fuzzy topology nor a generalization of Smarandache’s neutrosophic topology. Recently, Salama and Alblowi proposed a new definition of neutrosophic topology, that generalizes Çoker’s intuitionistic fuzzy topology. Here, we study this new definition and its relation to Smarandache’s paraconsistent neutrosophic sets., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Applied to a knot K of S1 in S3, Zeeman's n-twist-spinning construction produces a knot Kn of S2 in S4 [ E. C. Zeeman , Trans. Amer. Math. Soc. 115 (1965), 471–495;]. Here the author gives an explicit "movie presentation'' of Kn, that is, a description of Kn in terms of the cross-sections Kn∩S3×{t}., Comité Conjunto Hispano-Norteamericano, NSF, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Ibero-American Conference on Topology and its Applications (CITA-2012), For a locally compact group G we consider the class G-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G-invariant metric. We show that each X∈G-M admits a compatible G-invariant metric whose closed unit balls are small subsets of X. This is a key result to prove that X admits a closed equivariant embedding into an invariant convex subset V of a Banach G-space L such that L∖{0}∈G-M and V is a G-absolute extensor for the class G-M. On this way we establish two equivariant embedding results for proper G-spaces which may be considered as equivariant versions of the well-known Kuratowski–Wojdyslawski theorem and Arens–Eells theorem, respectively., CONACYT (Mexico)., CONACYT (Mexico)., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

In this paper the authors study irregular metacyclic branched covering spaces. These arise as follows: Suppose G is a Z/m extension of Z/n. Then G contains a cyclic subgroup of order m, Cm, which we suppose is not normal. Suppose G acts on a PL manifold X. Then there are maps X→X/Cm→X/G. The map M=X/Cm→X/G=S is an irregular metacyclic covering. (It is not induced by the action of a group on M because Cm is not normal.) In the most interesting case S is a simply connected manifold, often a sphere. Then M→S is a covering space in the usual sense away from a codimension 2 subset of S, called the branch set. Usually the branch set is taken to be a knot or link in S. Regular branched coverings have been widely studied in many contexts (knot theory, algebraic geometry, etc.), but irregular coverings have been much less studied although they are important also. For example, every closed oriented 3-manifold is a 3-fold irregular covering of S3 with branch set a knot. (This is the case G equals the dihedral group of order 6.). That this result does not generalize to dihedral groups of order 2p, p an odd prime, follows from Theorem 2 of this paper, which is too technical to state here. But specializing Theorem 2 to the context of rational homology and dihedral groups of order 2p, p an odd prime, the authors obtain the following theorem: dimH1(M;Q)≡0 modulo 1/2(p−1). The methods of proof in the paper are algebraic., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

For a topological space X, F(X) denotes the algebra of real-valued functions over X and C(X) the subalgebra of all functions in F(X) which are continuous. In this paper we characterize the uniformly dense linear subspaces of C(X) by means of the so-called "Lebesgue chain condition". This condition is a generalization to the unbounded case of the S-separation by Blasco and Molto for the bounded case. Through the Lebesgue chain condition we also characterize the linear subspaces of F(X) whose uniform closure is closed under composition with uniformly continuous functions., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A compact stable attractor K of a continuous flow on a locally compact metric space is shape equivalent to a compact positively invariant neighbourhood P in its basin of attraction, the shape equivalence being induced by the inclusion map. In the present article a simple counterexample is given to show that ∂K is in general not shape equivalent to ∂P , and an inquiry is set afoot as to the circumstances under which ∂K is shape equivalent to or shape dominates ∂P ., DGI, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Dedicated to Professor Jose M. Montesinos on the occasion of his 65th birthday, Let U ⊂ R3 be an open set and f : U → f(U) ⊂ R3 be a homeomorphism. Let p ∈ U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixedpoint indices of the iterates of f at p, (i(fn, p))n1, is, in general, unbounded. The main goal of this paper is to show that when {p} is an isolated invariant set, the sequence (i(fn, p))n1 is periodic. Conversely, we show that, for any periodic sequence of integers (In)n1 satisfying Dold’s necessary congruences, there exists an orientation-preserving homeomorphism such that i(fn, p) = In for every n 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p., MICINN, ANR, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

On every real Banach space X we introduce a locally convex topology tau(p), canonically associated to the weak-polynomial topology w(P). It is proved that tau(p) is the finest locally convex topology on X which is coarser than w(P). Furthermore, the convergence of sequences is considered, and sufficient conditions on X are obtained under which the convergent sequences for w(P) and for tau(P) either coincide with the weakly convergent sequences (when X has the Dunford-Pettis property) or coincide with the norm-convergent sequences (when X has nontrivial type)., DGES (Spain), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub