Morsova teorija in Heegaardov razcep 3-mnogoterosti : delo diplomskega seminarja

Peteh, Jan (author), Strle, Sašo (mentor)

V tem diplomskem delu predstavimo Heegaardov razcep sklenjenih 3-mnogoterosti, torej njihov razcep na ročajnike. Njegov obstoj, namesto s pomočjo triangulacij in Moisovega izreka, dokažemo s pomočjo Morsove teorije. Spoznamo Morsovo lemo, fundamentalna izreka Morsove teorije in njihove posledice, ki jih v nadaljevanju uporabimo za konstrukcijo funkcije, ki nam da ekspliciten Heegaardov razcep. Za konec si ogledamo še primer takega razcepa 3-dimenzionalnega torusa.

• Language: Slovenian
• Keywords: gladka mnogoterost, ročajnik, Morsova teorija, Heegaardov razcep, kobordizem
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-158906
• UDC: 515.1
• COBISS.SI-ID: 199740931
• Publication date in RUL: 22.06.2024

PETEH, Jan, 2024, Morsova teorija in Heegaardov razcep 3-mnogoterosti : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 26 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=158906

Secondary language

• Language: English
• Title: Morse theory and Heegaard splittings

Abstract

We present the notion of a Heegaard splitting of a closed manifold, which is defined as a decomposition into two handlebodies. A common proof of its existence follows the manifold's triangulability property and a theorem by Moise. Instead, we prove it using Morse theory. We take a look at Morse's lemma, fundamental theorems of Morse theory and their corollaries, which we then use in construction of a function that explicitly gives the Heegaard splitting. Lastly, we find such a splitting of a 3-dimensional torus.

• Keywords: smooth manifold, handlebody, Morse theory, Heegaard splitting, cobordism