Abstract In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it “Core fundamental groupoid” and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., “f: M → N is a homeomorphism if and only if f # M → 1 N is a topological groupoid isomorphism”. This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid.

Resumen Un espacio topológico X es C-normal si existe una función biyectiva f : X → Y , para algún espacio normal Y , tal que la restricción f ↾C: C → f(C) es un homeomorfismo para cada compacto C ⊂ X. El propósito de este trabajo es extender las clases conocidas de los espacios C-normales y aclarar el comportamiento de C-normalidad bajo varias operaciones topológicas habituales; en particular, se demuestra que la normalidad C no se conserva bajo subespacios cerrados, uniones, imágenes continuas y cerradas e imágenes inversas bajo funciones perfectas. Estos resultados se utilizan para responder algunas preguntas planteadas en [1], [2] y [6].

Abstract A topological space X is C-normal if there exists a bijective function f: X → Y , for some normal space Y , such that the restriction f ↾C: C → f(C) is a homeomorphism for each compact C ⊂ X. The purpose of this work is to extend the known classes of C-normal spaces and clarify the behavior of C-normality under several usual topological operations; in particular, it is proved that C-normality is not preserved under closed subspaces, unions, continuous and closed images, and inverse images under perfect functions. These results are used to answer some questions raised in [1], [2] and [6].

Abstract The theoretical existence of totally degenerate Kleinian groups is originally due to Bers and Maskit. In fact, Maskit proved that for any co-compact non-triangle Fuchsian group acting on the hyperbolic plane ℍ² there is a totally degenerate Kleinian group algebraically isomorphic to it. In this paper, by making a subtle modification to Maskit’s construction, we show that for any non-Euclidean crystallographic group F, such that ℍ²/F is not homeomorphic to a pant, there exists an extended Kleinian group G which is algebraically isomorphic to F and whose orientation-preserving half is a totally degenerate Kleinian group. Moreover, such an isomorphism is provided by conjugation by an orientation-preserving homeomorphism ϕ : ℍ² → Ω, where Ω is the region of discontinuity of G. In particular, this also provides another proof to Miyachi’s existence of totally degenerate finitely generated Kleinian groups whose limit set contains arcs of Euclidean circles.

Resumen La existencia teórica de grupos Kleinianos totalmente degenerados se debe originalmente a Bers y Maskit. De hecho, Maskit demostró que para cualquier grupo Fuchsiano co-compacto y no-triangular actuando en el plano hiperbólico ℍ² existe un grupo Kleiniano totalmente degenerado algebraicamente isomorfo a él. En este artículo, haciendo una modificación sutil a la construcción de Maskit, mostramos que para cualquier grupo cristalográfico no-Euclidiano F tal que ℍ²/F no es homeomorfo a un pantalón, existe un grupo Kleiniano extendido G que es algebraicamente isomorfo a F y cuya mitad que preserva orientación es un grupo Kleiniano totalmente degenerado. Más aún, un tal isomorfismo está dado por la conjugación por un homeomorfismo que preserva orientación ϕ: ℍ² → Ω, donde Ω es la región de discontinuidad de G. En particular, esto también entrega otra demostración del resultado de Miyachi acerca de la existencia de grupos Kleinianos totalmente degenerados finitamente generados cuyo conjunto límite contiene arcos de circunferencias Euclidianas.

Abstract: This paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy (4), we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh (12), and the converse of the main theorem found in (6). Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh's metric (12), define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated.

Resumen: Este artículo contribuye al estudio de los sistemas dinámicos simplécticos topológicos, y por tanto a la extensión de los sistemas dinámicos simplécticos suaves. Usando el resultado de la positividad de la energía de desplazamiento simpléctico (4), demostramos que cualquier generador de una isotopía simpléctica determina esta última. Esto entrega un análogo simpléctico de un resultado demostrado por Oh (12), y el inverso del teorema principal encontrado en (6). También entregamos herramientas para definir y estudiar los sistemas dinámicos simplécticos topológicos: construimos una métrica invariante por derecha en el grupo de homeomorfismos fuertemente simplécticos cuya restricción al grupo de homeomorfismos Hamiltonianos es equivalente a la métrica de Oh (12), definimos los análogos topológicos de la energía de desplazamiento simpléctico usual para conjuntos no-vacíos, y demostramos que esta última es positiva. Planteamos varios problemas abiertos.

Resumen. Alejandro Illanes preguntó si el pseudoarco P tiene hiperespacio segundo producto simétrico F2(P) único, es decir: si X es un continuo para el cual existe un homeomorfismoh : F2(P)) → F2(X), entonces, ¿es X homeomorfo al pseudoarco? En este trabajo probamos que si X es un continuo indescomponible y encadenable y Y es un continuo tal que F2(Y) es homeo- morfo a F2(X), entonces Y es indescomponible.

Abstract. Alejandro Illanes asked if the pseudoarc P has unique second symmetric product F2(P), this is, if X is a continuum such that there is a homeomorphism h : F2(P) → F2(X), then, is X homeomorphic to the pseudoarc? In this paper we show that if X is an indecomposable chainable continuum and Y is a continuum such that F2(Y) is homeomorphic to F2(X), then Y is indecomposable.

Un logro notable de la topología algorítmica es el resultado de A.A. Márkov sobre la insolubilidad del problema del homeomorfismo para variedades. Posteriormente, Boone, Haken y Poénaru extendieron la idea original de Márkov al caso de variedades suaves cerradas. Una primera dificultad era la introducción de una representación finita de una variedad diferenciable o combinatórica que la describiese de forma natural. En este trabajo extendemos dicha representación a variedades suaves compactas y proponemos una definición de variedad suave representable

A remarkable achievement of algorithmic topology is A.A.Markov's theorem on the unsolvability of the homeomorphism problem for manifolds. Boone, Haken and Poénaru extended Markov's original proof to the case of closed smooth manifolds. One of their initial difficulties was the introduction of a natural finite representation of a differentiable and/or combinatorial manifold. In this paper we extend this representation to compact smooth manifolds and propose an extension to smooth manifolds

Sea H un espacio de Hilbert real (o complejo). Todo operador no negativo L ∈ L(H) admite una única raíz cuadrada no negativa R ∈ L(H), esto es, un operador no negativo R ∈ L(H) tal que R² = L. Sea <img width=54 height=19 src="img/revistas/rein/v33n1/v33n1a02e2.jpg">el conjunto de los isomorfismos no negativos en L(H). Primero probaremos que <img width=54 height=19 src="img/revistas/rein/v33n1/v33n1a02e2.jpg">es una variedad de Banach (real). Denotando como L½ la raíz cuadrada no negativa de L, en [3] Richard Bouldin prueba que L½ depende continuamente de L (esta prueba es no trivial). Este resultado tiene varias aplicaciones. Por ejemplo, es usado para encontrar la descomposición polar de un operador limitado. Esta descomposición polar nos lleva a determinar los subespacios espectrales positivos y negativos de cualquier operador autoadjunto, y además, lleva a definir el índice de Máslov. El autor de este artículo da una prueba alternativa (y un poco más simplificada) de que L½ depende continuamente de L, y además, prueba que la aplicación <img width=159 height=38 src="img/revistas/rein/v33n1/v33n1a02e3.jpg"> es un homeomorfismo

Let H be a real (or complex) Hilbert space. Every nonnegative operator L ∈ L(H) admits a unique nonnegative square root R ∈ L(H), i.e., a nonnegative operator R ∈ L(H) such that R² = L. let <img width=54 height=19 src="img/revistas/rein/v33n1/v33n1a02e2.jpg">be the set of nonnegative isomorphisms in L(H). First we will show that <img width=54 height=19 src="img/revistas/rein/v33n1/v33n1a02e2.jpg">is a convex (real) Banach manifold. Denoting by L½ the nonnegative square root of L. In [3], Richard Bouldin proves that L½ depends continuously on L (this proof is nontrivial). This result has several applications. For example, it is used to find the polar decomposition of a bounded operator. This polar decomposition allows us to determine the positive and negative spectral subespaces of any selfadjoint operator, and moreover, allows us to define the Maslov index. The autor of the paper under review provides an alternative proof (and a little more simplified) that L½ depends continuously on L, and moreover, he shows that the map <img width=159 height=38 src="img/revistas/rein/v33n1/v33n1a02e3.jpg"> is a homeomorphism

We give a proof of the estimate (1.1) which is the main ingredient in the proof that the set SSympeo (M, ω) of strong symplectic homeomorphisms of a compact symplectic manifold (M, ω) forms a group [1].

Probamos la estimación (1.1) que es el principal elemento en la demostración que el conjuntos SSympeo (M, ω) de homeomorfismos simplécticos fuertes de una variedad simpléctica compacta (M, ω) genera un grupo [1].

Resumen. Un continuo es un espacio métrico, compacto y conexo. Un continuo X es homogéneo si para cualesquiera dos de sus puntos x1 y x2 de X, existe un homeomorfismo h: X <img width=18 height=12 src="img/revistas/rein/v29n2/v29n2a02f1.jpg">X tal que h(x1) = x2. Presentaremos un poco de historia, ejemplos y propiedades de este tipo de continuos. Daremos una demostración del Teorema de descomposición aposindética de Jones.

Abstract. A continuum is a compact, connected, metric space. A continuum X is homogeneous provided that for each pair of points x1 and x2 of X, there exists a homeomorphism h: X <img width=18 height=12 src="img/revistas/rein/v29n2/v29n2a02f1.jpg">X such that h(x1) = x2. We present a bit of history, examples and properties of this kind of continua. We give a proof of Jones's Aposyndetic Decomposition Theorem.

In this paper, we first introduce a new class of closed map called ĝ-closed map. Moreover, we introduce a new class of homeomorphism called ĝ-homeomorphism, which are weaker than homeomorphism. We prove that gc-homeomorphism and ĝ-homeomorphism are independent. We also introduce ĝ*-homeomorphisms and prove that the set of all ĝ*-homeomorphisms forms a group under the operation of composition of maps.

Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In this article we present a new approach which uses fields of directions defined on triangulations(FDTs) to treat both the problem of getting an homeomorphism between the models and that of morphing them. Consider that an scaled version(P1 ) of one of the objects, has already been adequately placed in the interior of the other(P2. ) The objective of the first part of the approach, is to obtain a field of unit vectors defined on a triangulation of the space between P1 and P2. This field must have no singularities and the trajectories determined by it will be later used to get warping and morphing transformations between P1 and P2. The morphing transformations obtained have the good property of being topology preserving ones but it can be hard to get an FDT defined on a triangulation of P1 - P2 and the intermediate models can have a very large number of faces. To illustrate those aspects, transformations between simple models are presented