El estudio del infinito a través del espacio exterior
In this thesis we introduce the notion of exterior space to study the properties of the spaces at infinite. One of the main problems of the proper homotopy is that there are few limits and colimits to satisfy CM1 axiom of Quillen closed model category. The category we define, the exterior spaces cat...
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Universidad de La Rioja (España)
1998
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Categoría de modelos cerrada de Quillen espacio exterior grupos de homotopía de Brown-Grossman grupos de homotopía de Steenrod teorema de Whitehead homotopía propia sucesión exacta larga de homotopía asociada a un morfismo categoría localizada Quillen closed model category Exterior space Whitehead theorem Brown-Grossman homotopy groups Steenrod homotopy groups Proper homotopy long exact homotopy sequence associated to a morphism localised category |
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Categoría de modelos cerrada de Quillen espacio exterior grupos de homotopía de Brown-Grossman grupos de homotopía de Steenrod teorema de Whitehead homotopía propia sucesión exacta larga de homotopía asociada a un morfismo categoría localizada Quillen closed model category Exterior space Whitehead theorem Brown-Grossman homotopy groups Steenrod homotopy groups Proper homotopy long exact homotopy sequence associated to a morphism localised category García Pinillos, Mónica El estudio del infinito a través del espacio exterior |
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In this thesis we introduce the notion of exterior space to study the properties of the spaces at infinite.
One of the main problems of the proper homotopy is that there are few limits and colimits to satisfy CM1 axiom of Quillen closed model category. The category we define, the exterior spaces category, provides a nice framework. The proper category can be considered as a full subcategory of the category of the exterior spaces. Moreover, the category of the exterior spaces satisfies all the axioms of closed model category. This way, we can study proper category by taking advantage of the techniques and results available in this axiomatic model. In particular, CM1, will allow to develop different homotopy constructions throughout finite limits and colimits.
One of the main problems of the proper category is that, until the moment, some homotopy constructions as homotopy fibres or loop spaces can't be developed. The utilization of the exterior spaces allows the construction of homotopy fibres and loop spaces. In this work we have obtained a consecutive homotopy fibres sequence and loop spaces to study groups type Brown-Grossman and other sequence to study groups type Steenrod
The category of the exterior spaces consists of exterior spaces as objects and exterior maps as morphisms. More in detail an exterior space is a topological space, (X, Tx), enriched with an additional structure we name externology. The definition of externology is suggested by the properties of the neighbourhoods at infinite (complements of closed compact sets) that will form the externology. An externology is a non empty collection of open subsets of X, we name them exterior open subsets, satisfying that the finite intersection of exterior open subsets is an exterior open subset and that if an exterior open subset is contained in an open subset, then the open subset is also an exterior open subset.
An externology is a topology but the reciprocal it is not true.
A map is said to be exterior if it is continuous respect to the topology and the externology.
One of the main results is Theorem 2.1.6. In this result we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Brown -Grossman homotopy groups, that we denominate exterior homotopy groups. We name this structure exterior as well as its associated notions.
We also obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base succession and, as a particular case, the long exact sequence of a pair in the category of pairs the exterior spaces with exterior base sequence.
In this thesis we also define the notion of N-complex. An N-complex is constructed inductively. The n-skeleton is constructed from the n-1 skeleton by gluing N-cells of dimension n, that are sequences of n-spheres (n greater or equals to zero) with exterior maps such as if has the weak topology and externology respect to n-skeletons filtration. We prove that N-complexes are exterior cofibrant objects. We prove a Whitehead theorem for N-complex and, as a particular case, a proper Whitehead theorem.
We can also prove that it is possible to provide the category of the exterior spaces with other structure of closed model category, we will denominate cylindric, as well as its associated notions to differentiate from the ones defined before for the exterior structure of closed model category and that were named exterior. In other important result, Theorem 5.2.5., we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Steenrod homotopy groups, that we denominate cylindric homotopy groups.
Likewise we obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base ray and, as a particular case, the long exact sequence of a pair in the category of pairs of the exterior spaces with exterior base ray.
In an analogous way to N-complexes, we define R-complexes, which play a similar role to CW-complexes in standard homotopy. We prove a Whitehead theorem in terms of cylindric groups and other results.
Finally, we compare the closed model categories induced in the category of the exterior spaces with exterior base ray by the exterior and cylindric structures. The main result establishes an equivalence of categories, restricted to cofibrant cylindric objects, between the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the cylindric model category structure, and the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the exterior model category structure. |
| author2 |
Hernández Paricio, Luis Javier (Universidad de La Rioja) |
| author_facet |
Hernández Paricio, Luis Javier (Universidad de La Rioja) García Pinillos, Mónica |
| format |
text (thesis) |
| author |
García Pinillos, Mónica |
| author_sort |
García Pinillos, Mónica |
| title |
El estudio del infinito a través del espacio exterior |
| title_short |
El estudio del infinito a través del espacio exterior |
| title_full |
El estudio del infinito a través del espacio exterior |
| title_fullStr |
El estudio del infinito a través del espacio exterior |
| title_full_unstemmed |
El estudio del infinito a través del espacio exterior |
| title_sort |
el estudio del infinito a través del espacio exterior |
| publisher |
Universidad de La Rioja (España) |
| publishDate |
1998 |
| url |
https://dialnet.unirioja.es/servlet/oaites?codigo=28 |
| work_keys_str_mv |
AT garciapinillosmonica elestudiodelinfinitoatravesdelespacioexterior |
| _version_ |
1718346593760968704 |
| spelling |
oai-TES00000023682016-04-13El estudio del infinito a través del espacio exteriorGarcía Pinillos, MónicaCategoría de modelos cerrada de Quillenespacio exteriorgrupos de homotopía de Brown-Grossmangrupos de homotopía de Steenrodteorema de Whiteheadhomotopía propiasucesión exacta larga de homotopía asociada a un morfismocategoría localizadaQuillen closed model categoryExterior spaceWhitehead theoremBrown-Grossman homotopy groupsSteenrod homotopy groupsProper homotopylong exact homotopy sequence associated to a morphismlocalised categoryIn this thesis we introduce the notion of exterior space to study the properties of the spaces at infinite. One of the main problems of the proper homotopy is that there are few limits and colimits to satisfy CM1 axiom of Quillen closed model category. The category we define, the exterior spaces category, provides a nice framework. The proper category can be considered as a full subcategory of the category of the exterior spaces. Moreover, the category of the exterior spaces satisfies all the axioms of closed model category. This way, we can study proper category by taking advantage of the techniques and results available in this axiomatic model. In particular, CM1, will allow to develop different homotopy constructions throughout finite limits and colimits. One of the main problems of the proper category is that, until the moment, some homotopy constructions as homotopy fibres or loop spaces can't be developed. The utilization of the exterior spaces allows the construction of homotopy fibres and loop spaces. In this work we have obtained a consecutive homotopy fibres sequence and loop spaces to study groups type Brown-Grossman and other sequence to study groups type Steenrod The category of the exterior spaces consists of exterior spaces as objects and exterior maps as morphisms. More in detail an exterior space is a topological space, (X, Tx), enriched with an additional structure we name externology. The definition of externology is suggested by the properties of the neighbourhoods at infinite (complements of closed compact sets) that will form the externology. An externology is a non empty collection of open subsets of X, we name them exterior open subsets, satisfying that the finite intersection of exterior open subsets is an exterior open subset and that if an exterior open subset is contained in an open subset, then the open subset is also an exterior open subset. An externology is a topology but the reciprocal it is not true. A map is said to be exterior if it is continuous respect to the topology and the externology. One of the main results is Theorem 2.1.6. In this result we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Brown -Grossman homotopy groups, that we denominate exterior homotopy groups. We name this structure exterior as well as its associated notions. We also obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base succession and, as a particular case, the long exact sequence of a pair in the category of pairs the exterior spaces with exterior base sequence. In this thesis we also define the notion of N-complex. An N-complex is constructed inductively. The n-skeleton is constructed from the n-1 skeleton by gluing N-cells of dimension n, that are sequences of n-spheres (n greater or equals to zero) with exterior maps such as if has the weak topology and externology respect to n-skeletons filtration. We prove that N-complexes are exterior cofibrant objects. We prove a Whitehead theorem for N-complex and, as a particular case, a proper Whitehead theorem. We can also prove that it is possible to provide the category of the exterior spaces with other structure of closed model category, we will denominate cylindric, as well as its associated notions to differentiate from the ones defined before for the exterior structure of closed model category and that were named exterior. In other important result, Theorem 5.2.5., we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Steenrod homotopy groups, that we denominate cylindric homotopy groups. Likewise we obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base ray and, as a particular case, the long exact sequence of a pair in the category of pairs of the exterior spaces with exterior base ray. In an analogous way to N-complexes, we define R-complexes, which play a similar role to CW-complexes in standard homotopy. We prove a Whitehead theorem in terms of cylindric groups and other results. Finally, we compare the closed model categories induced in the category of the exterior spaces with exterior base ray by the exterior and cylindric structures. The main result establishes an equivalence of categories, restricted to cofibrant cylindric objects, between the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the cylindric model category structure, and the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the exterior model category structure.En esta memoria se introduce la noción de espacio exterior para el estudio de las propiedades de los espacios en el infinito. Una de las desventajas de la categoría propia es que no posee suficientes límites y colímites para satisfacer el axioma CM1 de categoría de modelos cerrada de Quillen. La categoría que definimos en esta memoria, la de los espacios exteriores, contiene a la categoría propia y satisface todos los axiomas de la categoría de modelos cerrada. De esta forma, se puede estudiar la categoría propia aprovechando las técnicas y resultados de los que disponemos en ese modelo axiomático. En particular, CM1, permitirá realizar diversas construcciones homotópicas a través de los límites y colímites finitos. Uno de los problemas de la categoría propia es que, hasta el momento, no se habían podido construir adecuadas fibras homotópicas y adecuados espacios de lazos. La utilización de los espacios exteriores ha permitido la construcción de fibras homotópicas y espacios de lazos. En esta memoria se ha obtenido una sucesión de fibras homotópicas consecutivas para el estudio de los grupos de tipo Brown - Grossman y otra sucesión para los grupos de tipo Steenrod. La categoría de los espacios exteriores es aquella cuyos objetos son espacios exteriores y cuyos morfismos son las aplicaciones exteriores. Más en detalle, un espacio exterior es un espacio topológico (X, Tx) enriquecido con una estructura adicional, que llamamos externología. La definición de externología viene sugerida por las propiedades de los entornos del infinito (complementos de los conjuntos que sean cerrados y compactos) que constituirán una externología. Una externología es una familia no vacía de abiertos de , que denominaremos abiertos externos, verificando que la intersección finita de abiertos externos es también un abierto externo y tal que si un abierto externo está contenido en un abierto, el abierto es también un abierto externo. Una externología es una topología pero no ocurre que una externología sea una topología. Una aplicación se dice exterior si es continua respecto a la topología y respecto a la externología. Uno de los resultados claves de la memoria es el Teorema 2.1.6, en el que probamos que la categoría de los espacios exteriores tiene una estructura de categoría de modelos cerrada inducida por las equivalencias débiles determinadas por los grupos de tipo Brown-Grossman que en la memoria denominamos grupos de homotopía exterior. Obtenemos también una sucesión exacta larga de un morfismo en la categoría de los espacios exteriores con sucesión base exterior y, como caso particular, la de un par en la categoría de los pares de espacios exteriores con sucesión base exterior. En esta memoria se define la noción de N-complejo. Un N-complejo se construye de forma inductiva. El n-esqueleto se construye a partir del n-1 esqueleto pegando N-celdas de dimensión n, sucesiones exteriores de n-esferas para n mayor o igual que cero, mediante aplicaciones exteriores y de forma que tenga la topología y externología débiles respecto a la filtración de n- esqueletos. Se prueba que todos los N-complejos son objetos cofibrantes exteriores. Demostramos un teorema de Whitehead para N-complejos y, como caso particular, un teorema de Whitehead propio. Por otro lado, se demuestra que es posible dar a la categoría de los espacios exteriores otra estructura de categoría de modelos cerrada, que denominaremos cilíndrica así como a sus nociones asociadas, para distinguirlas de las anteriores que denominamos exteriores. En el Teorema 5.2.5. se demuestra la existencia de otra estructura diferente, que ahora está inducida por los grupos de homotopía tipo Steenrod, que llamamos cilíndricos. Asimismo obtenemos una sucesión exacta larga de un morfismo en la categoría de los espacios exteriores con rayo base exterior y, como caso particular, la de la de un par en la categoría de los pares de espacios exteriores con rayo base exterior. De forma análoga a los N-complejos, se definen los R-complejos, que desempeñan un papel similar a los CW-complejos en homotopía estándar. Probamos un teorema de Whitehead en términos de grupos cilíndricos, entre otros resultados. Para finalizar, se realiza una comparación entre las estructuras de categoría de modelos cerrada inducidas en la categoría de los espacios exteriores con rayo base exterior por las estructuras exterior y cilíndrica. El resultado principal establece una equivalencia de categorías restringida a los objetos cofibrados cilíndricos entre la categoría localizada de la categoría de los espacios exteriores con rayo base exterior, obtenida al invertir por equivalencias débiles, con la estructura de categoría de modelos cilíndrica, y la categoría localizada de la categoría de los espacios exteriores con rayo base exterior obtenida al invertir por equivalencias débiles con la estructura de categoría de modelos exterior.Universidad de La Rioja (España)Hernández Paricio, Luis Javier (Universidad de La Rioja)1998text (thesis)application/pdfhttps://dialnet.unirioja.es/servlet/oaites?codigo=28spaLICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. 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