Jose Calcines - Academia.edu (original) (raw)
Papers by Jose Calcines
arXiv (Cornell University), Dec 22, 2011
In this paper we introduce, by means of the category of exterior spaces and using a process that ... more In this paper we introduce, by means of the category of exterior spaces and using a process that generalizes the Alexandroff compactification, an analogue notion of numerable covering of a space in the proper and exterior setting. An application is given for fibrewise proper homotopy equivalences.
arXiv (Cornell University), May 17, 2019
We give a characterization of relative category in the sense of Doeraene-El Haouari by using open... more We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.
arXiv (Cornell University), Jun 24, 2011
In this paper we analyze some relationships between the topological complexity of a space X and t... more In this paper we analyze some relationships between the topological complexity of a space X and the category of C∆ X , the homotopy cofibre of the diagonal map ∆X : X → X × X. We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
arXiv (Cornell University), Mar 18, 2021
The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is... more The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fibrewise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally we give sufficient conditions so that both notions agree.
arXiv (Cornell University), Nov 2, 2007
In this article a sequential theory in the category of spaces and proper maps is described and de... more In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
arXiv (Cornell University), Feb 15, 2013
We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibre... more We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions.
Glasnik Matematicki, Nov 13, 2017
In this paper we analyse some applications of the category of exterior spaces to the study of dyn... more In this paper we analyse some applications of the category of exterior spaces to the study of dynamical systems (flows). The limit space and end space of an exterior space are used to construct different types of limit spaces and end spaces of a dynamical system. In this work we analyse the relationships between the notions and constructions given by the exterior structures of a continuous flow and the more usual notions of omega-limits, first prolongational limits and several types of almost periodic points (Poisson-stable points, non-wandering points) of a flow.
Homology, Homotopy and Applications, 2002
Well-known techniques from homological algebra and algebraic topology allow one to construct a co... more Well-known techniques from homological algebra and algebraic topology allow one to construct a cohomology theory for groups on which the action of a fixed group is given. After a brief discussion on the modules to be considered as coefficients, the first section of this paper is devoted to providing some definitions for this cohomology theory and then to proving that they are all equivalent. The second section is mainly dedicated to summarizing certain properties of this equivariant group cohomology and to showing several relationships with the ordinary group cohomology theory.
Graphical representations for the homogeneous bivariate Newton’s method
Applied Mathematics and Computation, 2015
In this paper we propose a new and effective strategy to apply Newton's method to the problem... more In this paper we propose a new and effective strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid some numerical difficulties, such as divisions by values close to zero. In fact, we consider an iteration map defined on a real augmented projective plane. So, we obtain a global description of the basins of attraction of the fixed points associated to the intersection of the curves. As an application of our techniques, we can plot the basins of attraction of the roots in the following geometric models: hemisphere, hemicube, Mobius band, square and disk. We can also give local graphical representations on any rectangle of the plane.
arXiv (Cornell University), Jul 24, 2023
In this paper, we provide sufficient conditions for a space X to satisfy the Ganea conjecture for... more In this paper, we provide sufficient conditions for a space X to satisfy the Ganea conjecture for topological complexity. To achieve this, we employ two auxiliary invariants: weak topological complexity in the sense of Berstein-Hilton, along with a certain stable version of it. Several examples are discussed.
Mathematische Zeitschrift, Jul 28, 2012
In this paper we analyze some relationships between the topological complexity of a space X and t... more In this paper we analyze some relationships between the topological complexity of a space X and the category of C∆ X , the homotopy cofibre of the diagonal map ∆X : X → X × X. We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
arXiv (Cornell University), Jul 25, 2023
In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented ... more In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented semi-simplicial sets. These constructions are obtained as particular cases of a certain action from a co-semi-simplicial set on an augmented semi-simplicial set. We also consider cylinders and subdivision operators in the algebraic setting of augmented sequences of integers. These operators are defined either by taking an action of matrices on sequences of integers (using binomial matrices) or by taking the simple product of sequences and matrices. We compare both the geometric and algebraic contexts using the sequential cardinal functor | • |, which associates the augmented sequence |X| = (|Xn|) n≥−1 to each augmented semi-simplicial finite set X. Here, |Xn| stands for the finite cardinality of the set of n-simplices Xn. The sequential cardinal functor transforms the action of any co-semi-simplicial set into the action of a matrix on a sequence. Therefore, we can easily calculate the number of simplices of cylinders or subdivisions of an augmented semi-simplicial set. Alternatively, instead of using the action of a matrix on a sequence, we can also compute suitable matrices and consider the product of an augmented sequence of integers and an infinite augmented matrix of integers. The calculation of these matrices is related mainly to binomial, chain-power-set, and Stirling numbers. From another point of view, these matrices can be considered as continuous automorphisms of the Baer-Specker topological group.
Applied Categorical Structures, Jun 1, 2004
The notion of exterior space consists of a topological space together with a certain nonempty fam... more The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity' while an exterior map is a continuous map which is 'continuous at infinity'. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T, T } of suitable exterior spaces. For this goal, for a given exterior space T , we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown-Grossman,Čerin-Steenrod, p-cylindrical, Baues-Quintero and Farrell-Taylor-Wagoner groups, as well as the standard (Hurewicz) homotopy groups. The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T, T } , it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.
In this paper we introduce, by means of the category of exterior spaces and using a process that ... more In this paper we introduce, by means of the category of exterior spaces and using a process that generalizes the Alexandroff compactification, an analogue notion of numerable covering of a space in the proper and exterior setting. An application is given for fibrewise proper homotopy equivalences.
Topology and its Applications
We give a characterization of relative category in the sense of Doeraene-El Haouari by using open... more We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.
Bulletin of the Belgian Mathematical Society - Simon Stevin
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From ... more We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
Bulletin of the Belgian Mathematical Society - Simon Stevin
In this paper we analyze some applications of the category of exterior spaces to the study of dyn... more In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future part" of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space is used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces L r (X) andL r (X) are constructed and their relations with the subflows of periodic, Poisson stable points and Ω r-limits of X are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point a. When a runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.
Homology, Homotopy and Applications
An exterior space is a topological space provided with a quasifilter of open subsets (closed by f... more An exterior space is a topological space provided with a quasifilter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod,Čech, Brown-Grossman). On the other hand, for a discrete semi-flow, one can analyze fixed points, periodic points, omega-limits, et cetera. In this paper, we introduce a notion of exterior discrete semi-flow, which is a mix of exterior space and discrete semi-flow. We see that a discrete semi-flow can be provided with the structure of an exterior discrete semi-flow by taking as structure of exterior space the family of right-absorbing open subsets, which can be used to study the relation between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end points are used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points. We also analyze the exterior discrete semi-flow structure induced by the family of open neighborhoods of a given sub-semi-flow.
Revista De La Academia De Ciencias Exactas Fisicas Quimicas Y Naturales De Zaragoza, 2002
The notion of exterior space consists of a topological space together with a certain nonempty fam... more The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. In this paper we present and develop the category of exterior spaces as a good framework for proper homotopy theory. As an application we give a new version of the Whitehead theorem for proper homotopy theory. We also give simplicial models for this new category and we analyze singular and realization-type functors for these models. 1 Introducción Uno de los problemas existentes en la categoría de los espacios y aplicaciones propias, P, es que no verifica la axiomática de Quillen (que exige tener límites y colímites finitos) pues la categoría P no tiene objeto final y en general no existen push-outs. Una forma de establecer un marco de trabajo en teoría de homotopía propia es escoger axiomáticas menos restrictivas, como la noción de categoría cofibrada. En este sentido se demuestra que P tiene estructura de categoría cofibrada [7]. Existen otras posibilidades, por ejemplo, se puede encajar P en una categoría completa y cocompleta y usar teorías de homotopía que asuman la existencia de límites y colímites. En esta dirección, se tiene el encaje de Edwards y Hastings de la subcategoría de P de los espacios σ-compactos, Hausdorff y localmente compactos en la categoría de homotopía de proespacios. Una desventaja de este encaje es la gran restricción hecha en P. Otro problema es que las contrucciones homotópicas producen proespacios que muchas veces no pueden ser interpretados geométricamente como espacios. García-Pinillos [14] propone en su tesis una nueva solución. Introduce la noción de espacio exterior, de forma que la categoría de los espacios exteriores, E, es completa y cocompleta y demuestra que P se puede considerar como una subcategoría plena suya. Además, E, tiene varias estructuras de categoría de modelos cerrada. Por otro lado, la obtención de modelos algebraicos para tipos de homotopía de espacios topológicos ha sido un objetivo primordial en la homotopía algebraica. Uno de los primeros modelos con información
Homotopía propia simplicial
arXiv (Cornell University), Dec 22, 2011
In this paper we introduce, by means of the category of exterior spaces and using a process that ... more In this paper we introduce, by means of the category of exterior spaces and using a process that generalizes the Alexandroff compactification, an analogue notion of numerable covering of a space in the proper and exterior setting. An application is given for fibrewise proper homotopy equivalences.
arXiv (Cornell University), May 17, 2019
We give a characterization of relative category in the sense of Doeraene-El Haouari by using open... more We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.
arXiv (Cornell University), Jun 24, 2011
In this paper we analyze some relationships between the topological complexity of a space X and t... more In this paper we analyze some relationships between the topological complexity of a space X and the category of C∆ X , the homotopy cofibre of the diagonal map ∆X : X → X × X. We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
arXiv (Cornell University), Mar 18, 2021
The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is... more The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fibrewise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally we give sufficient conditions so that both notions agree.
arXiv (Cornell University), Nov 2, 2007
In this article a sequential theory in the category of spaces and proper maps is described and de... more In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
arXiv (Cornell University), Feb 15, 2013
We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibre... more We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions.
Glasnik Matematicki, Nov 13, 2017
In this paper we analyse some applications of the category of exterior spaces to the study of dyn... more In this paper we analyse some applications of the category of exterior spaces to the study of dynamical systems (flows). The limit space and end space of an exterior space are used to construct different types of limit spaces and end spaces of a dynamical system. In this work we analyse the relationships between the notions and constructions given by the exterior structures of a continuous flow and the more usual notions of omega-limits, first prolongational limits and several types of almost periodic points (Poisson-stable points, non-wandering points) of a flow.
Homology, Homotopy and Applications, 2002
Well-known techniques from homological algebra and algebraic topology allow one to construct a co... more Well-known techniques from homological algebra and algebraic topology allow one to construct a cohomology theory for groups on which the action of a fixed group is given. After a brief discussion on the modules to be considered as coefficients, the first section of this paper is devoted to providing some definitions for this cohomology theory and then to proving that they are all equivalent. The second section is mainly dedicated to summarizing certain properties of this equivariant group cohomology and to showing several relationships with the ordinary group cohomology theory.
Graphical representations for the homogeneous bivariate Newton’s method
Applied Mathematics and Computation, 2015
In this paper we propose a new and effective strategy to apply Newton's method to the problem... more In this paper we propose a new and effective strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid some numerical difficulties, such as divisions by values close to zero. In fact, we consider an iteration map defined on a real augmented projective plane. So, we obtain a global description of the basins of attraction of the fixed points associated to the intersection of the curves. As an application of our techniques, we can plot the basins of attraction of the roots in the following geometric models: hemisphere, hemicube, Mobius band, square and disk. We can also give local graphical representations on any rectangle of the plane.
arXiv (Cornell University), Jul 24, 2023
In this paper, we provide sufficient conditions for a space X to satisfy the Ganea conjecture for... more In this paper, we provide sufficient conditions for a space X to satisfy the Ganea conjecture for topological complexity. To achieve this, we employ two auxiliary invariants: weak topological complexity in the sense of Berstein-Hilton, along with a certain stable version of it. Several examples are discussed.
Mathematische Zeitschrift, Jul 28, 2012
In this paper we analyze some relationships between the topological complexity of a space X and t... more In this paper we analyze some relationships between the topological complexity of a space X and the category of C∆ X , the homotopy cofibre of the diagonal map ∆X : X → X × X. We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
arXiv (Cornell University), Jul 25, 2023
In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented ... more In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented semi-simplicial sets. These constructions are obtained as particular cases of a certain action from a co-semi-simplicial set on an augmented semi-simplicial set. We also consider cylinders and subdivision operators in the algebraic setting of augmented sequences of integers. These operators are defined either by taking an action of matrices on sequences of integers (using binomial matrices) or by taking the simple product of sequences and matrices. We compare both the geometric and algebraic contexts using the sequential cardinal functor | • |, which associates the augmented sequence |X| = (|Xn|) n≥−1 to each augmented semi-simplicial finite set X. Here, |Xn| stands for the finite cardinality of the set of n-simplices Xn. The sequential cardinal functor transforms the action of any co-semi-simplicial set into the action of a matrix on a sequence. Therefore, we can easily calculate the number of simplices of cylinders or subdivisions of an augmented semi-simplicial set. Alternatively, instead of using the action of a matrix on a sequence, we can also compute suitable matrices and consider the product of an augmented sequence of integers and an infinite augmented matrix of integers. The calculation of these matrices is related mainly to binomial, chain-power-set, and Stirling numbers. From another point of view, these matrices can be considered as continuous automorphisms of the Baer-Specker topological group.
Applied Categorical Structures, Jun 1, 2004
The notion of exterior space consists of a topological space together with a certain nonempty fam... more The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity' while an exterior map is a continuous map which is 'continuous at infinity'. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T, T } of suitable exterior spaces. For this goal, for a given exterior space T , we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown-Grossman,Čerin-Steenrod, p-cylindrical, Baues-Quintero and Farrell-Taylor-Wagoner groups, as well as the standard (Hurewicz) homotopy groups. The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T, T } , it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.
In this paper we introduce, by means of the category of exterior spaces and using a process that ... more In this paper we introduce, by means of the category of exterior spaces and using a process that generalizes the Alexandroff compactification, an analogue notion of numerable covering of a space in the proper and exterior setting. An application is given for fibrewise proper homotopy equivalences.
Topology and its Applications
We give a characterization of relative category in the sense of Doeraene-El Haouari by using open... more We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.
Bulletin of the Belgian Mathematical Society - Simon Stevin
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From ... more We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
Bulletin of the Belgian Mathematical Society - Simon Stevin
In this paper we analyze some applications of the category of exterior spaces to the study of dyn... more In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future part" of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space is used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces L r (X) andL r (X) are constructed and their relations with the subflows of periodic, Poisson stable points and Ω r-limits of X are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point a. When a runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.
Homology, Homotopy and Applications
An exterior space is a topological space provided with a quasifilter of open subsets (closed by f... more An exterior space is a topological space provided with a quasifilter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod,Čech, Brown-Grossman). On the other hand, for a discrete semi-flow, one can analyze fixed points, periodic points, omega-limits, et cetera. In this paper, we introduce a notion of exterior discrete semi-flow, which is a mix of exterior space and discrete semi-flow. We see that a discrete semi-flow can be provided with the structure of an exterior discrete semi-flow by taking as structure of exterior space the family of right-absorbing open subsets, which can be used to study the relation between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end points are used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points. We also analyze the exterior discrete semi-flow structure induced by the family of open neighborhoods of a given sub-semi-flow.
Revista De La Academia De Ciencias Exactas Fisicas Quimicas Y Naturales De Zaragoza, 2002
The notion of exterior space consists of a topological space together with a certain nonempty fam... more The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. In this paper we present and develop the category of exterior spaces as a good framework for proper homotopy theory. As an application we give a new version of the Whitehead theorem for proper homotopy theory. We also give simplicial models for this new category and we analyze singular and realization-type functors for these models. 1 Introducción Uno de los problemas existentes en la categoría de los espacios y aplicaciones propias, P, es que no verifica la axiomática de Quillen (que exige tener límites y colímites finitos) pues la categoría P no tiene objeto final y en general no existen push-outs. Una forma de establecer un marco de trabajo en teoría de homotopía propia es escoger axiomáticas menos restrictivas, como la noción de categoría cofibrada. En este sentido se demuestra que P tiene estructura de categoría cofibrada [7]. Existen otras posibilidades, por ejemplo, se puede encajar P en una categoría completa y cocompleta y usar teorías de homotopía que asuman la existencia de límites y colímites. En esta dirección, se tiene el encaje de Edwards y Hastings de la subcategoría de P de los espacios σ-compactos, Hausdorff y localmente compactos en la categoría de homotopía de proespacios. Una desventaja de este encaje es la gran restricción hecha en P. Otro problema es que las contrucciones homotópicas producen proespacios que muchas veces no pueden ser interpretados geométricamente como espacios. García-Pinillos [14] propone en su tesis una nueva solución. Introduce la noción de espacio exterior, de forma que la categoría de los espacios exteriores, E, es completa y cocompleta y demuestra que P se puede considerar como una subcategoría plena suya. Además, E, tiene varias estructuras de categoría de modelos cerrada. Por otro lado, la obtención de modelos algebraicos para tipos de homotopía de espacios topológicos ha sido un objetivo primordial en la homotopía algebraica. Uno de los primeros modelos con información
Homotopía propia simplicial