Closed injective systems and its fundamental limit spaces

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.Comment: 18 pages, 2 figure

The trivial homotopy class of maps from two-complexes into the real projective plane

We study reasons related to two-dimensional CW-complexes which prevent an extension of the Hopf--Whitney Classification Theorem for maps from those complexes into the real projective plane, even in the simpler situation in which the complex has trivial second integer cohomology group. We conclude that for such a two-complex $K$, the following assertions are equivalent: (1) Every based map from $K$ into the real projective plane is based homotopic to a constant map; (2) The skeleton pair $(K,K^1)$ is homotopy equivalent to that of a model two-complex induced by a balanced group presentation; (3) The number of two-dimensional cells of $K$ is equal to the first Betti number of its one-skeleton; (4) $K$ is acyclic; (5) Every based map from $K$ into the circle $S^1$ is based homotopic to a~constant map

Roots of maps from 2-dimensional complexes into closed surfaces

Este texto é resultado de um estudo detalhado da teoria topológica de raízes para aplicações de complexos CW 2-dimensionais em superfícies fechadas (compactas e sem bordo). Diversas abordagens dos problemas envolvidos nesta teoria são apresentadas, algumas inclusive bastante diferenciadas com respeito aos parâmetros da teoria clássicaThis text is the result of a detailed study of the topological root theory for maps from 2-dimensional CW complex into closed surfaces (compact and without boundary surfaces). Several approaches to the problems involved in this theory are presented, some of which are quite different with respect to the parameters of the classical theor

Minimizing roots of maps into the two-sphere

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

Coincidence of maps from two-complexes into graphs

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

Embeddings of cartesian products of spheres and its connected sums in codimension 1

Estudamos inicialmente resultados de classificação de difeomorfismos de produtos de esferas de mesma dimensão. Tratado isto, estudamos os mergulhos suaves de produtos de três esferas, sendo a primeira de dimensão um e as demais de dimensão maior ou igual a um, com a dimensão da última maior ou igual a da segunda, em uma esfera em codimensão um, e buscamos a total caracterização do fecho das duas componentes conexas do complementar de tais mergulhos. Tratamos com enfoque especial os mergulhos do produto de três esferas de dimensão um na esfera de dimensão quatro, e, finalmente, estudamos problemas de classificação de mergulhos PL localmente não-enodados de somas conexas de toros em codimensão um.We study initially results of classification of difeomorfisms of Cartesian products of spheres of same dimension. Treated this, we study the smooth embeddings of cartesian products of three spheres, being the first one of dimension one and excessively of bigger or equal dimension to one, with the dimension of the last equal greater or of second, in a sphere in codimension one, and search the total characterization of the latch of the two connected components of complementing of such embeddings. We deal with special approach the embeddings of the product to three spheres to dimension one in the sphere dimension four, and, finally, we study problems of classification of PL locally unknotted embeddings of connected sums of torus on codimension one

Strong surjectivity of maps from 2-complexes into the 2-sphere

Given a model 2-complex K(P) of a group presentation P, we associate to it an integer matrix Delta(P) and we prove that a cellular map f : K(P) -> S(2) is root free (is not strongly surjective) if and only if the diophantine linear system Delta(P) Y = (deg) over right arrow (f) has an integer solution, here (deg) over right arrow (f) is the so-called vector-degree of fFAPESP[2007/05843-5]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Minimal Nielsen Root Classes and Roots of Liftings

Given a continuous map f : K -> M from a 2-dimensional CW complex into a closed surface, the Nielsen root number N(f) and the minimal number of roots mu(f) of f satisfy N(f) <= mu(f). But, there is a number mu(C)(f) associated to each Nielsen root class of f, and an important problem is to know when mu(f) = mu(C)(f)N(f). In addition to investigate this problem, we determine a relationship between mu(f) and mu((f) over tilde), when (f) over tilde f is a lifting of f through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.FAPESP[2007/05843-5