{"id":{"repo_id":"dialnet","oai_identifier":"oai:dialnet.unirioja.es:TES0000023239"},"canonical_url":"https://search.dev.ndltd.org/etd/dialnet/oai:dialnet.unirioja.es:TES0000023239","repository":{"repo_id":"dialnet","name":"Dialnet","base_url":"https://dialnet.unirioja.es/oaites/OAIHandler"},"display":{"title":"Spectral systems: new instances and algorithms","abstract":"The classification problem in algebraic topology motivates the development of increasingly refined invariants capable of distinguishing complex spaces. Spectral sequences have long played a central role in this endeavor, providing structured approximations to homological and homotopical invariants. More recently, spectral systems, introduced by Benjamin Matschke, have emerged as a broad generalization of spectral sequences, allowing filtrations over more general indexing objects and offering greater flexibility in relating graded data to homology. In this thesis we investigate spectral systems both from a theoretical and a computational perspective. First of all, we construct and study a spectral system that combines Serre and Eilenberg--Moore spectral sequences, clarifying their interactions and identifying within this unified framework some preludes that were introduced by Neumann and Szymik. This demonstrates that spectral systems provide a natural setting for combining distinct spectral sequences and for revealing structural relations that are not visible at the level of classical constructions. To investigate a general source of spectral systems, we also develop a generalization of multicomplexes to higher-dimensional settings. Several possible notions of generalized multicomplexes are introduced and analyzed, together with their associated spectral systems. We study their behavior under standard algebraic operations and establish conditions under which effective homology can be obtained. These constructions are illustrated through explicit examples, such as towers of twisted Cartesian products and the effective homology of a bicomplex. Finally, we dualize Matschke's geometric approach for the Eilenberg--Moore generalized spectral sequence, providing a study of the homology of multi-factored fibered products of topological spaces. We address the coproduct problem for the Cobar construction, and we define an iterated Cobar for higher dimensions using generalized multicomplexes. In addition, we define a generalized filtration on this new Cobar chain complex that gives the Eilenberg--Moore spectral system. For this new spectral system, as well as for the previous ones, we present algorithms for their computation using the method of effective homology. This method, created by Rubio and Sergeraert, is a tool to compute the homology of complicated and possibly infinite spaces. Moreover, for the two spectral systems that we introduced first, we also present an implementation on the Kenzo system using effective homology techniques. With this, we extend the capabilities of the existing modules of Kenzo and provide practical tools for analysis and future research.","abstract_html":"The classification problem in algebraic topology motivates the development of increasingly refined invariants capable of distinguishing complex spaces. Spectral sequences have long played a central role in this endeavor, providing structured approximations to homological and homotopical invariants. More recently, spectral systems, introduced by Benjamin Matschke, have emerged as a broad generalization of spectral sequences, allowing filtrations over more general indexing objects and offering greater flexibility in relating graded data to homology. In this thesis we investigate spectral systems both from a theoretical and a computational perspective. First of all, we construct and study a spectral system that combines Serre and Eilenberg--Moore spectral sequences, clarifying their interactions and identifying within this unified framework some preludes that were introduced by Neumann and Szymik. This demonstrates that spectral systems provide a natural setting for combining distinct spectral sequences and for revealing structural relations that are not visible at the level of classical constructions. To investigate a general source of spectral systems, we also develop a generalization of multicomplexes to higher-dimensional settings. Several possible notions of generalized multicomplexes are introduced and analyzed, together with their associated spectral systems. We study their behavior under standard algebraic operations and establish conditions under which effective homology can be obtained. These constructions are illustrated through explicit examples, such as towers of twisted Cartesian products and the effective homology of a bicomplex. Finally, we dualize Matschke's geometric approach for the Eilenberg--Moore generalized spectral sequence, providing a study of the homology of multi-factored fibered products of topological spaces. We address the coproduct problem for the Cobar construction, and we define an iterated Cobar for higher dimensions using generalized multicomplexes. In addition, we define a generalized filtration on this new Cobar chain complex that gives the Eilenberg--Moore spectral system. For this new spectral system, as well as for the previous ones, we present algorithms for their computation using the method of effective homology. This method, created by Rubio and Sergeraert, is a tool to compute the homology of complicated and possibly infinite spaces. Moreover, for the two spectral systems that we introduced first, we also present an implementation on the Kenzo system using effective homology techniques. With this, we extend the capabilities of the existing modules of Kenzo and provide practical tools for analysis and future research.","abstract_has_math":false,"creators":["Miguel Treviño, Daniel"],"institution":"Universidad de La Rioja (España)","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Romero Ibáñez, Ana (null)","Guidolin, Andrea (null)"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026","date_published":"2026","updated_at":"2026-07-24T06:27:11Z","subjects":[],"languages":["eng"],"rights":["LICENCIA 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. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dialnet.unirioja.es/servlet/oaites?codigo=402493","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Romero Ibáñez, Ana (null)","Guidolin, Andrea (null)"]},{"key":"dc:creator","label":"Author","values":["Miguel Treviño, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2026"]},{"key":"dc:publisher","label":"Institution","values":["Universidad de La Rioja (España)"]},{"key":"dc:type","label":"Dc Type","values":["text (thesis)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["LICENCIA 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. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dialnet.unirioja.es/servlet/oaites?codigo=402493"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The classification problem in algebraic topology motivates the development of increasingly refined invariants capable of distinguishing complex spaces. Spectral sequences have long played a central role in this endeavor, providing structured approximations to homological and homotopical invariants. More recently, spectral systems, introduced by Benjamin Matschke, have emerged as a broad generalization of spectral sequences, allowing filtrations over more general indexing objects and offering greater flexibility in relating graded data to homology. In this thesis we investigate spectral systems both from a theoretical and a computational perspective. First of all, we construct and study a spectral system that combines Serre and Eilenberg--Moore spectral sequences, clarifying their interactions and identifying within this unified framework some preludes that were introduced by Neumann and Szymik. This demonstrates that spectral systems provide a natural setting for combining distinct spectral sequences and for revealing structural relations that are not visible at the level of classical constructions. To investigate a general source of spectral systems, we also develop a generalization of multicomplexes to higher-dimensional settings. Several possible notions of generalized multicomplexes are introduced and analyzed, together with their associated spectral systems. We study their behavior under standard algebraic operations and establish conditions under which effective homology can be obtained. These constructions are illustrated through explicit examples, such as towers of twisted Cartesian products and the effective homology of a bicomplex. Finally, we dualize Matschke's geometric approach for the Eilenberg--Moore generalized spectral sequence, providing a study of the homology of multi-factored fibered products of topological spaces. We address the coproduct problem for the Cobar construction, and we define an iterated Cobar for higher dimensions using generalized multicomplexes. In addition, we define a generalized filtration on this new Cobar chain complex that gives the Eilenberg--Moore spectral system. For this new spectral system, as well as for the previous ones, we present algorithms for their computation using the method of effective homology. This method, created by Rubio and Sergeraert, is a tool to compute the homology of complicated and possibly infinite spaces. Moreover, for the two spectral systems that we introduced first, we also present an implementation on the Kenzo system using effective homology techniques. With this, we extend the capabilities of the existing modules of Kenzo and provide practical tools for analysis and future research.","El problema de clasificación en topología algebraica motiva el desarrollo de invariantes cada vez más refinados para distinguir espacios topológicos complicados. Las sucesiones espectrales han jugado un rol central en este aspecto, proporcionando una aproximación estructurada y gradual hacia la homología o la homotopía de un espacio. Más recientemente, los sistemas espectrales, introducidos por Benjamin Matschke, han emergido como una amplia generalización de las sucesiones espectrales, permitiendo filtraciones sobre conjuntos de índices más generales y ofreciendo una mayor flexibilidad a la hora de relacionar la información filtrada con la homología. En esta tesis investigamos los sistemas espectrales tanto desde un punto de vista teórico como práctico. En primer lugar, construimos y estudiamos un sistema espectral que combina las sucesiones espectrales de Serre y de Eilenberg--Moore, mostrando sus interacciones e identificando los preludios de Neumann y Szymik en este contexto. Esto demuestra que los sistemas espectrales son una estructura natural para combinar diferentes sucesiones espectrales, así como para mostrar relaciones estructurales que no son visibles al nivel de las construcciones clásicas. Para buscar una fuente general de sistemas espectrales, también desarrollamos una generalización de los multicomplejos a dimensiones más altas. Varias opciones posibles para esta generalización son introducidas y analizadas, junto con sus sistemas espectrales asociados. Estudiamos su comportamiento frente a operaciones algebraicas estándar, y establecemos condiciones bajo las cuales podemos obtener su homología efectiva. Estas construcciones serán ilustradas mediante ejemplos explícitos, como las torres de productos Cartesianos torcidos o la homología efectiva de un bicomplejo. Finalmente, dualizamos el enfoque geométrico de Matschke para la sucesión espectral generalizada de Eilenberg--Moore, proporcionando un estudio de la homología de productos fibrados de varios factores. Atacamos el problema del coproducto para la construcción Cobar, y definimos una Cobar iterada para dimensiones altas usando multicomplejos generalizados. Además, definimos una filtración generalizada sobre este nuevo complejo de cadenas Cobar, la cual tiene asociado el sistema espectral de Eilenberg--Moore. Para este nuevo sistema espectral, así como para los sistemas espectrales anteriores, presentamos algoritmos usando el método de la homología efectiva. Este método, definido por Rubio y Sergeraert, es una herramienta para calcular la homología de espacios complicados y posiblemente infinitos. Además, para nuestros primeros sistemas espectrales, presentamos una implementación de dichos algoritmos en el sistema Kenzo. Con esto, extendemos las capacidades de los módulos de Kenzo existentes y obtenemos herramientas prácticas para el análisis y la investigación futuros."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Spectral systems: new instances and algorithms"]}]}],"canonical_facts":{"dc:contributor":["Romero Ibáñez, Ana (null)","Guidolin, Andrea (null)"],"dc:creator":["Miguel Treviño, Daniel"],"dc:date":["2026"],"dc:description":["The classification problem in algebraic topology motivates the development of increasingly refined invariants capable of distinguishing complex spaces. Spectral sequences have long played a central role in this endeavor, providing structured approximations to homological and homotopical invariants. More recently, spectral systems, introduced by Benjamin Matschke, have emerged as a broad generalization of spectral sequences, allowing filtrations over more general indexing objects and offering greater flexibility in relating graded data to homology. In this thesis we investigate spectral systems both from a theoretical and a computational perspective. First of all, we construct and study a spectral system that combines Serre and Eilenberg--Moore spectral sequences, clarifying their interactions and identifying within this unified framework some preludes that were introduced by Neumann and Szymik. This demonstrates that spectral systems provide a natural setting for combining distinct spectral sequences and for revealing structural relations that are not visible at the level of classical constructions. To investigate a general source of spectral systems, we also develop a generalization of multicomplexes to higher-dimensional settings. Several possible notions of generalized multicomplexes are introduced and analyzed, together with their associated spectral systems. We study their behavior under standard algebraic operations and establish conditions under which effective homology can be obtained. These constructions are illustrated through explicit examples, such as towers of twisted Cartesian products and the effective homology of a bicomplex. Finally, we dualize Matschke's geometric approach for the Eilenberg--Moore generalized spectral sequence, providing a study of the homology of multi-factored fibered products of topological spaces. We address the coproduct problem for the Cobar construction, and we define an iterated Cobar for higher dimensions using generalized multicomplexes. In addition, we define a generalized filtration on this new Cobar chain complex that gives the Eilenberg--Moore spectral system. For this new spectral system, as well as for the previous ones, we present algorithms for their computation using the method of effective homology. This method, created by Rubio and Sergeraert, is a tool to compute the homology of complicated and possibly infinite spaces. Moreover, for the two spectral systems that we introduced first, we also present an implementation on the Kenzo system using effective homology techniques. With this, we extend the capabilities of the existing modules of Kenzo and provide practical tools for analysis and future research.","El problema de clasificación en topología algebraica motiva el desarrollo de invariantes cada vez más refinados para distinguir espacios topológicos complicados. Las sucesiones espectrales han jugado un rol central en este aspecto, proporcionando una aproximación estructurada y gradual hacia la homología o la homotopía de un espacio. Más recientemente, los sistemas espectrales, introducidos por Benjamin Matschke, han emergido como una amplia generalización de las sucesiones espectrales, permitiendo filtraciones sobre conjuntos de índices más generales y ofreciendo una mayor flexibilidad a la hora de relacionar la información filtrada con la homología. En esta tesis investigamos los sistemas espectrales tanto desde un punto de vista teórico como práctico. En primer lugar, construimos y estudiamos un sistema espectral que combina las sucesiones espectrales de Serre y de Eilenberg--Moore, mostrando sus interacciones e identificando los preludios de Neumann y Szymik en este contexto. Esto demuestra que los sistemas espectrales son una estructura natural para combinar diferentes sucesiones espectrales, así como para mostrar relaciones estructurales que no son visibles al nivel de las construcciones clásicas. Para buscar una fuente general de sistemas espectrales, también desarrollamos una generalización de los multicomplejos a dimensiones más altas. Varias opciones posibles para esta generalización son introducidas y analizadas, junto con sus sistemas espectrales asociados. Estudiamos su comportamiento frente a operaciones algebraicas estándar, y establecemos condiciones bajo las cuales podemos obtener su homología efectiva. Estas construcciones serán ilustradas mediante ejemplos explícitos, como las torres de productos Cartesianos torcidos o la homología efectiva de un bicomplejo. Finalmente, dualizamos el enfoque geométrico de Matschke para la sucesión espectral generalizada de Eilenberg--Moore, proporcionando un estudio de la homología de productos fibrados de varios factores. Atacamos el problema del coproducto para la construcción Cobar, y definimos una Cobar iterada para dimensiones altas usando multicomplejos generalizados. Además, definimos una filtración generalizada sobre este nuevo complejo de cadenas Cobar, la cual tiene asociado el sistema espectral de Eilenberg--Moore. Para este nuevo sistema espectral, así como para los sistemas espectrales anteriores, presentamos algoritmos usando el método de la homología efectiva. Este método, definido por Rubio y Sergeraert, es una herramienta para calcular la homología de espacios complicados y posiblemente infinitos. Además, para nuestros primeros sistemas espectrales, presentamos una implementación de dichos algoritmos en el sistema Kenzo. Con esto, extendemos las capacidades de los módulos de Kenzo existentes y obtenemos herramientas prácticas para el análisis y la investigación futuros."],"dc:format":["application/pdf"],"dc:identifier":["https://dialnet.unirioja.es/servlet/oaites?codigo=402493"],"dc:language":["eng"],"dc:publisher":["Universidad de La Rioja (España)"],"dc:rights":["LICENCIA 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. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"],"dc:title":["Spectral systems: new instances and algorithms"],"dc:type":["text (thesis)"]},"updated_at":"2026-07-24T06:27:11Z"}