Publication: Hyperspaces, shape theory and computational topology
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2016-08-16
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Universidad Complutense de Madrid
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Abstract
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 homotopía de la variedad se puede recuperar utilizando un conjunto finito de puntos (suficientemente denso) para el complejo de Vietoris-Rips. Los resultados de Petersen [58], comparando la distancia Gromov-Hausdorff de los compactos métricos con su tipo de homotopía, son también interesantes. Aquí, los poliedros salen a relucir en las demostraciones, no en los resultados...
This thesis is about approximations of metric compacta. The approximation and reconstruction of topological spaces using simpler ones is an old theme in geometric topology. One would like to construct a very simple space as similar as possible to the original space. Since it is very difficult (or does not make sense) to obtain a homeomorphic copy, the goal will be to find an space preserving some (algebraic) topological properties such as compactness, connectedness, separation axioms, homotopy type, homotopy and homology groups, etc. The first candidates to act as the simple spaces reproducing some properties of the original space are polyhedra. See the survey [45] for the main results. In the very beginnings of this idea, we must recall the studies of Alexandroff around 1920, relating the dimension of compact metric spaces with dimension of polyhedra by means of maps with controlled (in terms of distance) images or preimages. In a more modern framework, the idea of approximation can be carried out constructing a simplicial complex, based on our space, such as the Vietoris-Rips complex or the Cech complex, and compare its realization with it. In this direction, for example, we find the classical Nerve Lemma [12, 21] which claims that for a “good enough" open cover of the space (meaning an open covering with contractible or empty members and intersections), the nerve of the cover has the homotopy type of our original space. The problem is to find those good covers (if they exist). For Riemannian manifolds, there are some results concerning its approximation by means of the Vietoris-Rips complex. Hausmann showed [35] that the realization of the Vietoris-Rips complex of the manifold, for a small enough parameter choice, has the homotopy type of the manifold. In [40], Latschev proved a conjecture made by Hausmann: The homotopy type of the manifold can be recovered using only a (dense enough) finite set of points of it, for the Vietoris-Rips complex. The results of Petersen [58], comparing the Gromov-Hausdorff distance of metric compacta with their homotopy types, are also interesting. Here, polyhedra are just used in the proofs, not in the results...
This thesis is about approximations of metric compacta. The approximation and reconstruction of topological spaces using simpler ones is an old theme in geometric topology. One would like to construct a very simple space as similar as possible to the original space. Since it is very difficult (or does not make sense) to obtain a homeomorphic copy, the goal will be to find an space preserving some (algebraic) topological properties such as compactness, connectedness, separation axioms, homotopy type, homotopy and homology groups, etc. The first candidates to act as the simple spaces reproducing some properties of the original space are polyhedra. See the survey [45] for the main results. In the very beginnings of this idea, we must recall the studies of Alexandroff around 1920, relating the dimension of compact metric spaces with dimension of polyhedra by means of maps with controlled (in terms of distance) images or preimages. In a more modern framework, the idea of approximation can be carried out constructing a simplicial complex, based on our space, such as the Vietoris-Rips complex or the Cech complex, and compare its realization with it. In this direction, for example, we find the classical Nerve Lemma [12, 21] which claims that for a “good enough" open cover of the space (meaning an open covering with contractible or empty members and intersections), the nerve of the cover has the homotopy type of our original space. The problem is to find those good covers (if they exist). For Riemannian manifolds, there are some results concerning its approximation by means of the Vietoris-Rips complex. Hausmann showed [35] that the realization of the Vietoris-Rips complex of the manifold, for a small enough parameter choice, has the homotopy type of the manifold. In [40], Latschev proved a conjecture made by Hausmann: The homotopy type of the manifold can be recovered using only a (dense enough) finite set of points of it, for the Vietoris-Rips complex. The results of Petersen [58], comparing the Gromov-Hausdorff distance of metric compacta with their homotopy types, are also interesting. Here, polyhedra are just used in the proofs, not in the results...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología, leída el 15-12-2015