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Relative homotopy invariants of the type of the Lusternik-Schnirelmann category
Fasso Velenik, Agnese

HaupttitelRelative homotopy invariants of the type of the Lusternik-Schnirelmann category
TitelvarianteRelative Homotopie-Invarianten des Types der Kategorie von Lusternik-Schnirelmann
AutorFasso Velenik, Agnese
Geburtsort: Bologna, Italien
GutachterProf. Dr. Hans Scheerer
weitere GutachterProf. Dr. Yves Félix
Freie SchlagwörterLS-category, cone-length, sectional category, joins, rational invariants55P05, 55P62
DDC510 Mathematik
Zusammenfassung

Among the numerous homotopy invariants the category of Lusternik-Schnirelmann , or LS-category, 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 cone-length of a space. Moreover LS-category can be extended to continuous maps in three different ways, thus generating the F-category, the R-category and the LS-category of a map, which are analogous to the sectional category of a fibration. Finally Félix and Halperin gave a new dimension to the LS-category 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 F-category of a map.

In this thesis we are particularly interested in relative invariants of the type of the LS-category, such as F-category, R-category, LS-category, sectional category and cone-length of a map. In chapter 1 we introduce a few tools which are very useful to define the various relative categories: homotopy push-outs, homotopy pull-backs 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 LS-category and cone-length. In particular we give three equivalent definitions of the LS-category: in terms of coverings, of fat wedges and of Ganea maps, constructed by taking consecutive joins. We also give bounds for the LS-category and the cone-length of a product of spaces. On the other hand we introduce the F-category, the R-category and the LS-category of maps, giving for each of them three equivalent definitions, as well as the cone-length of a map.

In chapter 4 we find a bound for the cone-length of a product of maps and use it to obtain bounds for the F-category, the R-category and the LS-category 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 LS-category and of the F-category and their characterization directly in the rational context. We then introduce a rationalization of the R-category and the relative LS-category 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 R-category 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 R-category. In this case as for the classical LS-category 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/EinrichtungFB Mathematik und Informatik
Erscheinungsjahr2003
Dokumententyp/-SammlungenDissertation
Medientyp/FormatText
SpracheEnglisch
RechteNutzungsbedingungen
Tag der Disputation09.05.2003
Erstellt am04.11.2003 - 00:00:00
Letzte Änderung19.02.2010 - 12:40:34
 
Alte Darwin URLhttp://www.diss.fu-berlin.de/2003/277/
Statische URLhttp://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000001150
URNurn:nbn:de:kobv:188-2003002778
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