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Relative homotopy invariants of the type of the LusternikSchnirelmann category Fasso Velenik, Agnese 
Haupttitel  Relative homotopy invariants of the type of the LusternikSchnirelmann category 
Titelvariante  Relative HomotopieInvarianten des Types der Kategorie von LusternikSchnirelmann 
Autor  Fasso Velenik, Agnese
Geburtsort: Bologna, Italien 
Gutachter  Prof. Dr. Hans Scheerer 
weitere Gutachter  Prof. Dr. Yves Félix 
Freie Schlagwörter  LScategory, conelength, sectional category, joins, rational invariants55P05, 55P62 
DDC  510 Mathematik 
Zusammenfassung  Among the numerous homotopy invariants the category of LusternikSchnirelmann , or LScategory, of a topological space has aroused much interest since its definition in 1934. For example it was shown that it is related to another invariant: the conelength of a space. Moreover LScategory can be extended to continuous maps in three different ways, thus generating the Fcategory, the Rcategory and the LScategory of a map, which are analogous to the sectional category of a fibration. Finally Félix and Halperin gave a new dimension to the LScategory by transferring it into the context of rational homotopy theory: they gave a method to compute its rationalization directly in the category of commutative cochain algebras (in short: cca's). They also rationalized the Fcategory of a map. In this thesis we are particularly interested in relative invariants of the type of the LScategory, such as Fcategory, Rcategory, LScategory, sectional category and conelength of a map. In chapter 1 we introduce a few tools which are very useful to define the various relative categories: homotopy pushouts, homotopy pullbacks and joins. Then we give a brief description of rational homotopy theory in chapter 2: we state the equivalence of categories underlying it which links topological spaces and commutative cochain algebras (in short: cca's). We also define (relative) Sullivan algebras, which are particularly nice to deal with, and can be used as building blocks when modelizing some topological constructions such as joins. Chapter 3 is devoted on the one hand to a description of the original LScategory and conelength. In particular we give three equivalent definitions of the LScategory: in terms of coverings, of fat wedges and of Ganea maps, constructed by taking consecutive joins. We also give bounds for the LScategory and the conelength of a product of spaces. On the other hand we introduce the Fcategory, the Rcategory and the LScategory of maps, giving for each of them three equivalent definitions, as well as the conelength of a map. In chapter 4 we find a bound for the conelength of a product of maps and use it to obtain bounds for the Fcategory, the Rcategory and the LScategory of a product of maps. Chapter 5 contains a summary of part of a paper from Félix and Halperin giving a rationalization of the absolute LScategory and of the Fcategory and their characterization directly in the rational context. We then introduce a rationalization of the Rcategory and the relative LScategory and we state our main theorem, allowing to compute them directly in the cca setting. We give a proof of this assertion in chapter 6 by defining Ganea algebras and Ganea morphisms modelling Ganea spaces and maps. Some applications of the main theorem are given in chapter 7: we show that the Rcategory can take up any value, and we simplify our main result in case the map being considered is the inclusion of a fibre. Moreover we prove that the rational relative category of a spherical fibration does not depend only on the order of its Euler class as it is the case for its rational sectional category. Finally we devote our last chapter to the study of a new homotopy invariant: the sectional category of a sequence of maps, which generalizes both the sectional category of a fibration and the Rcategory. In this case as for the classical LScategory we give three equivalent definitions in terms of coverings, of generalized fat wedges and of generalized Ganea spaces. Moreover we rationalize the new invariant and prove a theorem allowing its direct computation in the rational setting. 
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Fachbereich/Einrichtung  FB Mathematik und Informatik 
Erscheinungsjahr  2003 
Dokumententyp/Sammlungen  Dissertation 
Medientyp/Format  Text 
Sprache  Englisch 
Rechte  Nutzungsbedingungen 
Tag der Disputation  09.05.2003 
Erstellt am  04.11.2003  00:00:00 
Letzte Änderung  19.02.2010  12:40:34 
Alte Darwin URL  http://www.diss.fuberlin.de/2003/277/ 
Statische URL  http://www.diss.fuberlin.de/diss/receive/FUDISS_thesis_000000001150 
URN  urn:nbn:de:kobv:1882003002778 
Zugriffsstatistik  