German Title: Funktionen mit einer minimalen Anzahl kritischen Punkten

PDF, English
Download (593kB)  Terms of use 
Abstract
In 1934 Lusternik and Schnirelmann introduced a new numerical topological invariant (the LusternikSchnirelmann category) during research into the global calculus of variation. They showed that this invariant carries important information about both the existence of critical points and the cardinality of the critical set. Further results on LusternikSchnirelmann category distract the attention from the original problem: finding crit(M) the minimal number of critical points of a smooth function on a given smooth manifold M. Only Cornea in 1998 analyzed the notion of crit, originally introduced by Takens in 1968. Inspired by Takens' paper and by Morse theory in the Smale setting we have analyzed the crit for products of manifolds. Using concepts from the graph theory we have found upper bounds of crit for products of special manifolds with generalized tori. The proofs of these results are based on the fusing lemma, which establishes sufficient conditions to construct a triad function with at most one critical point from a triad function with two critical points. In order to compute the crit for products of lens spaces with spheres we prove that the ball category is a lower bound for the number of the critical points of a function. The thesis contains also a paragraph about crit of manifolds with boundary.
Translation of abstract (English)
In 1934 Lusternik and Schnirelmann introduced a new numerical topological invariant (the LusternikSchnirelmann category) during research into the global calculus of variation. They showed that this invariant carries important information about both the existence of critical points and the cardinality of the critical set. Further results on LusternikSchnirelmann category distract the attention from the original problem: finding crit(M) the minimal number of critical points of a smooth function on a given smooth manifold M. Only Cornea in 1998 analyzed the notion of crit, originally introduced by Takens in 1968. Inspired by Takens' paper and by Morse theory in the Smale setting we have analyzed the crit for products of manifolds. Using concepts from the graph theory we have found upper bounds of crit for products of special manifolds with generalized tori. The proofs of these results are based on the fusing lemma, which establishes sufficient conditions to construct a triad function with at most one critical point from a triad function with two critical points. In order to compute the crit for products of lens spaces with spheres we prove that the ball category is a lower bound for the number of the critical points of a function. The thesis contains also a paragraph about crit of manifolds with boundary.
Item Type:  Dissertation 

Supervisor:  Dold, Prof. Dr. Albrecht 
Date of thesis defense:  16 March 2001 
Date Deposited:  29 Nov 2001 00:00 
Date:  2000 
Faculties / Institutes:  The Faculty of Mathematics and Computer Science > Department of Mathematics 
Subjects:  510 Mathematics 
Controlled Keywords:  MorseTheorie, LjusternikSnirel'manKategorie, Gerichteter Graph 
Uncontrolled Keywords:  Critical Points , hcobordism , Oriented Trees 