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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

This work is about analytic invariants of isolated hypersurface singularities and combinatorial invariants of numerical semigroups. The first part deals with analytic and topological invariants of an isolated hypersurface singularity. Our main contributions are the following: first we provide a closed formula for the minimal Tjurina number in an equisingularity class of a plane branch in terms of topological invariants of the branch, secondly we address a question of Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of an isolated plane curve singularity; we extend this question to isolated surface singularities in C3 which gives the clue to provide a complete answer to Dimca and Greuel's question. Moreover, we show the connection of the extended question with an old standing conjecture posed by Durfee. Finally, we establish K. Saito's continuous limit distribution for the spectrum of Newton non-degenerate isolated hypersurface singularities and link this problem with our generalization of Dimca and Greuel's question. As a consequence, this provides a new way of understanding the important role of Durfee's conjecture in the context of isolated hypersurface singularities...

We show that the homotopy type of any connected component of the contactomorphism groupof a tight contact 3-manifold is characterized by the homotopy type of the di↵eomorphismgroup plus some data provided by the topology of the formal contactomorphism space. As aconsequence, we show that every connected component of the space of Legendrian long knotsin R3 has the homotopy type of the underlying smooth long knot space. This implies that anyconnected component of the space of Legendrian embeddings in S3 is homotopy equivalentto the space K(G, 1) ⇥ U(2), with G computed by A. Hatcher and R. Budney. Similarstatements are proven for Legendrian embeddings in R3 and for transverse embeddings inS3. We compute the homotopy type of the contactomorphism group of several tight 3-folds:S1 ⇥ S2, Legendrian fibrations over compact orientable surfaces and finite quotients of thestandard 3-sphere. In fact, the computations show that the method works whenever we haveknowledge of the topology of the di↵eomorphism group. We prove several statements on theway that have interest by themselves: the computation of the homotopy groups of the space ofnon-parametrized Legendrians, a multiparametric convex surface theory, a characterizationof formal Legendrian simplicity in terms of the space of tight contact structures on thecomplement of a Legendrian and the existence of common trivializations for multi-parametricfamilies of tight R3. We also compare the situation with the overtwisted case. Finally, weapply contact topological methods to study the topology of the space of parametrized smoothknots in R4...

We have developed both a homology theory and a homotopy theory in the context of metric subanalytic germs (see Definition 2.1). The former is called MD homology and is covered in Chapter 2, which contains a paper that is joined work with my PhD advisors Javier Fernández de Bobadilla and María Pe Pereira and with Edson Sam-paio. The latter is called MD homotopy and is covered in Chapter 3. Both theories are functors from a category of germs of metric subanalytic spaces (resp. germs of metric subanalytic spaces that are punctured in a way that will be defined) to a category of commutative diagrams of groups. For the concrete definition of the domain categories see Definition 2.10 and Definition 3.47 respectively; for the target categories see Definition 2.42 and Definition 3.52 respectively. Similarly to classical homology and homotopy theories, the groups appearing in the target category are abelian in the homology theory for any degree and in the homotopy theory for degree n > 1...

Given a variety of contact M, a subvariety L is said Legendriana if it is tangent to the contact distribution and has maximum dimension (the maximum dimension is conditioned by the condition of being tangent to the contact distribution). Note by Leg (L) the space of Isotope Legendrianas subvarieties to L.Y. Eliashberg and L. Polterovich conducted a study on the manageability of the contactomorphism groups that allowed us to find a relationship between the orderliness of the Leg (L) space and the existence of positive Legendrian ties. This relationship is extensible to the universal Leg (L) coater as long as we consider that the previous ties are also contractile. Positive links of contactomorphisms are constructed in Legendrian subvarieties in different cases. In particular, we partially recover the result of G. Liu, who affirms that every Legendriana Loose variety admits a positive bond, assuming that certain little restrictive topological properties on the Legendriana subvariety are met. Moreover, the contractibility of ties is proved by assuming an extra topological property.

In this thesis we study geometric structures on orbifolds. Our main interest lies in the relationship between such structures in orbifolds and corresponding geometric structures of associated manifolds. One instance of this is the symplectic resolution of orbifold singularities in which we associate a symplectic manifold (the resolution) to a symplectic orbifold. Resolution of symplectic orbifolds is a natural extension to the symplectic category of the classical problem of resolution of singularities in algebraic geometry. Apart from the intrinsic interest of the problem of resolution of singularities in symplectic geometry, resolution of symplectic orbifolds also gives a powerful method to construct symplectic manifolds starting from symplectic orbifolds. With this ideain mind, we develop a method to resolve a certain type of symplectic orbifolds, which we call homegenous isotropy orbifolds. These do not cover orbifolds in full generality, but they suce to construct interesting manifolds...

Esta tesis está dedicada al estudio, utilizando técnicas topológicas, de la estructura cualitativa de un flujo cerca de un compacto invariante. Este fue uno delos primeros temas clásicos desarrollados por H. Poincaré, I. Bendixson, A. Andronovy S. Lefschetz al inicio de la teoría cualitativa de ecuaciones diferenciales,con contribuciones de autores como D.M. Grobman, P. Hartman, J.K. Hale y A.Stokes y muchos otros. Cabe destacar, por ejemplo, la descripción presentadapor T. Ura e I. Kimura en [97] o las teorías de índice de Wazewski y Conley que encapsulan, en un sentido topológico, propiedades del flujo cerca de un compacto invariante (aislado).La importancia del estudio de la estructura de un flujo cerca de un compacto invariante queda patente en las palabras de J. Auslander, N.P. Bhatia y P. Seibert[5] recogidas en el célebre artículo sobre el concepto de un atractor [60] de J.Milnor:“En el estudio de las propiedades topológicas de las ecuaciones diferenciales ordinarias, la teoría de estabilidad de compactos invariantes (que podrían ser considerados generalizaciones de puntos críticos y ciclos límite) juega un papel central”...

Esta tesis trata sobre aproximaciones de espacios métricos compactos. La aproximación y reconstrucción de espacios topológicos mediante otros más sencillos es un tema antigüo en topología geométrica. La idea es construir un espacio muy sencillo lo más parecido posible al espacio original. Como es muy difícil (o incluso no tiene sentido) intentar obtener una copia homeomorfa, el objetivo será encontrar un espacio que preserve algunas propriedades topológicas (algebraicas o no) como compacidad, conexión, axiomas de separación, tipo de homotopía, grupos de homotopía y homología, etc. Los primeros candidatos como espacios sencillos con propiedades del espacio original son los poliedros. Ver el artículo [45] para los resultados principales. En el germen de esta idea, destacamos los estudios de Alexandroff en los años 20, relacionando la dimensión del compacto métrico con la dimensión de ciertos poliedros a través de aplicaciones con imágenes o preimágenes controladas (en términos de distancias). En un contexto más moderno, la idea de aproximación puede ser realizada construyendo un complejo simplicial basado en el espacio original, como el complejo de Vietoris-Rips o el complejo de Cech y comparar su realización con él. En este sentido, tenemos el clásico lema del nervio [12, 21] el cual establece que para un recubrimiento por abiertos “suficientemente bueno" del espacio (es decir, un recubrimiento con miembros e intersecciones contractibles o vacías), el nervio del recubrimiento tiene el tipo de homotopía del espacio original. El problema es encontrar estos recubrimientos (si es que existen). Para variedades Riemannianas, existen algunos resultados en este sentido, utilizando los complejos de Vietoris-Rips. Hausmann demostró [35] que la realización del complejo de Vietoris-Rips de la variedad, para valores suficientemente bajos del parámetro, tiene el tipo de homotopía de dicha variedad. En [40], Latschev demostró una conjetura establecida por Hausmann: El tipo de homot

El sofista Gorgias, hace ya más de veinticinco siglos, dejó planteada la gran fractura en la que el hombre se desenvuelve, al mostrar la inconmensurabilidad entre las cosas, el conocimiento de las cosas y las manifestaciones lingüísticas de lo que son o creemos que son las cosas: pues nada existe, y si existiera no podría ser conocido, y aunque pudiera ser conocido sería incomunicable.Gran parte de la actividad filosófica se esfuerza en seleccionar materiales e inventar herramientas que permitan construir puentes adecuados para entrelazar las tres Ideas de referencia de Gorgias: Mundo, Mente y Lenguaje. Aquí trataremos la cuestión del lenguaje.