In this article we introduce the concept of limit space and fundamental limit space for the so-ca... more In this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. We present the main results on existence and uniqueness of limit spaces and several concrete examples. In the main section of the text, we show that the closed injective system can be considered as objects of a category whose morphisms are the so-called cis-morphisms. Moreover, the transition to fundamental limit space can be considered a functor from this category into category of topological spaces. Later, we show results about properties on functors and counter-functors for inductive closed injective system and fundamental limit spaces. We finish with the presentation of some results of characterization of fundamental limite space for some special systems and the study of so-called perfect properties.
For the model two-complex K of the group presentation P=〈 x,y | x^k+1yxy 〉, with k≥1 odd, we desc... more For the model two-complex K of the group presentation P=〈 x,y | x^k+1yxy 〉, with k≥1 odd, we describe representatives for all free and based homotopy classes of maps from K into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all k≥1 odd, H^2(K;mathbbZ)=0 and, for k≥3 odd, there exist maps from K into the real projective plane which are strongly surjective. For k=1, there is no such a strongly surjective map. 55N25, 57M20.
This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2-... more This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2-sphere. Given such a map $f:K\rightarrow S^2$ we define two integers $\zeta(f)$ and $\zeta(K,d_f)$, which are upper bounds for the minimal number of roots of $f$, denote be $\mu(f)$. The number $\zeta(f)$ is only defined when $f$ is a cellular map and $\zeta(K,d_f)$ is defined when $K$ is homotopy equivalent to the 2-sphere. When these two numbers are defined, we have the inequality $\mu(f)\leq\zeta(K,d_f)\leq\zeta(f)$, where $d_f$ is the so-called homological degree of $f$. We use these results to present two very interesting examples of maps from 2-complexes homotopy equivalent to the sphere into the sphere.
RESUMO – A ocorrencia de precipitacao e um dos fenomenos de maior importância na determinacao do ... more RESUMO – A ocorrencia de precipitacao e um dos fenomenos de maior importância na determinacao do clima e, como a maioria dos demais fenomenos climaticos, apresenta uma variabilidade espacial continua que pode ser detectada atraves de metodos geoestatisticos especiais. Conhecida a grande influencia do relevo topografico no clima especifico de cada regiao, e de se esperar que esta variavel apresente correlacoes espaciais com a ocorrencia de precipitacoes, e que o determinismo desta correlacao possa de alguma maneira auxiliar na elaboracao de conclusoes mais precisas que envolvam estes fenomenos. Neste trabalho analisou-se rigorosamente a variabilidade espacial da altitude e das precipitacoes pluviometricas, alem da correlacao existente entre estas variaveis. Concluiu-se que estas variaveis apresentam forte dependencia espacial e estao diretamente correlacionadas. O mapeamento da ocorrencia de ambos os fenomenos foi realizado tambem atraves de metodos geoestatisticos, baseando-se nas i...
Given a finite and connected two-dimensional CW -complex K with fundamental group Π and second in... more Given a finite and connected two-dimensional CW -complex K with fundamental group Π and second integer cohomology group H(K;Z) finite of odd order, we prove that: (1) for each local integer coefficient system α : Π → Aut(Z) over K, the corresponding twisted cohomology group H(K;αZ) is finite of odd order, we say order c (α), and there exists a natural function – which resemble that one defined by the twisted degree – from the set [K;RP]α of the based homotopy classes of based maps inducing α on π1 into H(K;αZ), which is a bijection; (2) the set [K;RP ]α of the (free) homotopy classes of based maps inducing α on π1 is finite of order c(α) = (c (α) + 1)/2; (3) all but one of the homotopy classes [f ] ∈ [K;RP]α are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism f : H(RP;̺Z) → H (K;αZ), where ̺ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups H(K;αZ) are provided for several...
For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \... more For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for all free and based homotopy classes of maps from $K$ into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all $k\geq1$ odd, $H^2(K;mathbb{Z})=0$ and, for $k\geq3$ odd, there exist maps from $K$ into the real projective plane which are strongly surjective. For $k=1$, there is no such a strongly surjective map. 55N25, 57M20.
Abstract We build a countable collection of two-dimensional CW complexes with trivial second inte... more Abstract We build a countable collection of two-dimensional CW complexes with trivial second integer cohomology group and, from each of them, a strong surjection onto the torus. Furthermore, we prove that such two-complexes are the simplest with these properties. This answers, for dimension two, a problem originally proposed in the 2000's for dimension three.
The main theorem of this article provides a necessary and sufficient condition for a pair of maps... more The main theorem of this article provides a necessary and sufficient condition for a pair of maps from a two-complex into a one-complex (a graph) can be homotoped to be coincidence free. As a consequence of it, we prove that a pair of maps from a two-complex into the circle can be homotoped to be coincidence free if and only if the two maps are homotopic. We also obtain an alternative proof for the known result that every pair of maps from a graph into the bouquet of a circle and an interval can be homotoped to be coincidence free. As applications of the main theorem, we characterize completely when a pair of maps from the bi-dimensional torus into the bouquet of a circle and an interval can be homotoped to be coincidence free, and we prove that every pair of maps from the Klein bottle into such a bouquet can be homotoped to be coincidence free.
We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed p... more We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.
For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the ro... more For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the root set by deforming the maps through homotopies. After presenting the classification of the homotopy classes of such maps, we prove that the minimal root set for a non null-homotopic map is either a circle or the disjoint union of two circle, according its range is $S^2$ or $\RP^2$, respectively.
In this article we introduce the concept of limit space and fundamental limit space for the so-ca... more In this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. We present the main results on existence and uniqueness of limit spaces and several concrete examples. In the main section of the text, we show that the closed injective system can be considered as objects of a category whose morphisms are the so-called cis-morphisms. Moreover, the transition to fundamental limit space can be considered a functor from this category into category of topological spaces. Later, we show results about properties on functors and counter-functors for inductive closed injective system and fundamental limit spaces. We finish with the presentation of some results of characterization of fundamental limite space for some special systems and the study of so-called perfect properties.
For the model two-complex K of the group presentation P=〈 x,y | x^k+1yxy 〉, with k≥1 odd, we desc... more For the model two-complex K of the group presentation P=〈 x,y | x^k+1yxy 〉, with k≥1 odd, we describe representatives for all free and based homotopy classes of maps from K into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all k≥1 odd, H^2(K;mathbbZ)=0 and, for k≥3 odd, there exist maps from K into the real projective plane which are strongly surjective. For k=1, there is no such a strongly surjective map. 55N25, 57M20.
This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2-... more This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2-sphere. Given such a map $f:K\rightarrow S^2$ we define two integers $\zeta(f)$ and $\zeta(K,d_f)$, which are upper bounds for the minimal number of roots of $f$, denote be $\mu(f)$. The number $\zeta(f)$ is only defined when $f$ is a cellular map and $\zeta(K,d_f)$ is defined when $K$ is homotopy equivalent to the 2-sphere. When these two numbers are defined, we have the inequality $\mu(f)\leq\zeta(K,d_f)\leq\zeta(f)$, where $d_f$ is the so-called homological degree of $f$. We use these results to present two very interesting examples of maps from 2-complexes homotopy equivalent to the sphere into the sphere.
RESUMO – A ocorrencia de precipitacao e um dos fenomenos de maior importância na determinacao do ... more RESUMO – A ocorrencia de precipitacao e um dos fenomenos de maior importância na determinacao do clima e, como a maioria dos demais fenomenos climaticos, apresenta uma variabilidade espacial continua que pode ser detectada atraves de metodos geoestatisticos especiais. Conhecida a grande influencia do relevo topografico no clima especifico de cada regiao, e de se esperar que esta variavel apresente correlacoes espaciais com a ocorrencia de precipitacoes, e que o determinismo desta correlacao possa de alguma maneira auxiliar na elaboracao de conclusoes mais precisas que envolvam estes fenomenos. Neste trabalho analisou-se rigorosamente a variabilidade espacial da altitude e das precipitacoes pluviometricas, alem da correlacao existente entre estas variaveis. Concluiu-se que estas variaveis apresentam forte dependencia espacial e estao diretamente correlacionadas. O mapeamento da ocorrencia de ambos os fenomenos foi realizado tambem atraves de metodos geoestatisticos, baseando-se nas i...
Given a finite and connected two-dimensional CW -complex K with fundamental group Π and second in... more Given a finite and connected two-dimensional CW -complex K with fundamental group Π and second integer cohomology group H(K;Z) finite of odd order, we prove that: (1) for each local integer coefficient system α : Π → Aut(Z) over K, the corresponding twisted cohomology group H(K;αZ) is finite of odd order, we say order c (α), and there exists a natural function – which resemble that one defined by the twisted degree – from the set [K;RP]α of the based homotopy classes of based maps inducing α on π1 into H(K;αZ), which is a bijection; (2) the set [K;RP ]α of the (free) homotopy classes of based maps inducing α on π1 is finite of order c(α) = (c (α) + 1)/2; (3) all but one of the homotopy classes [f ] ∈ [K;RP]α are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism f : H(RP;̺Z) → H (K;αZ), where ̺ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups H(K;αZ) are provided for several...
For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \... more For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for all free and based homotopy classes of maps from $K$ into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all $k\geq1$ odd, $H^2(K;mathbb{Z})=0$ and, for $k\geq3$ odd, there exist maps from $K$ into the real projective plane which are strongly surjective. For $k=1$, there is no such a strongly surjective map. 55N25, 57M20.
Abstract We build a countable collection of two-dimensional CW complexes with trivial second inte... more Abstract We build a countable collection of two-dimensional CW complexes with trivial second integer cohomology group and, from each of them, a strong surjection onto the torus. Furthermore, we prove that such two-complexes are the simplest with these properties. This answers, for dimension two, a problem originally proposed in the 2000's for dimension three.
The main theorem of this article provides a necessary and sufficient condition for a pair of maps... more The main theorem of this article provides a necessary and sufficient condition for a pair of maps from a two-complex into a one-complex (a graph) can be homotoped to be coincidence free. As a consequence of it, we prove that a pair of maps from a two-complex into the circle can be homotoped to be coincidence free if and only if the two maps are homotopic. We also obtain an alternative proof for the known result that every pair of maps from a graph into the bouquet of a circle and an interval can be homotoped to be coincidence free. As applications of the main theorem, we characterize completely when a pair of maps from the bi-dimensional torus into the bouquet of a circle and an interval can be homotoped to be coincidence free, and we prove that every pair of maps from the Klein bottle into such a bouquet can be homotoped to be coincidence free.
We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed p... more We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.
For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the ro... more For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the root set by deforming the maps through homotopies. After presenting the classification of the homotopy classes of such maps, we prove that the minimal root set for a non null-homotopic map is either a circle or the disjoint union of two circle, according its range is $S^2$ or $\RP^2$, respectively.
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Papers by Marcio Colombo Fenille