Difeomorfizmi ploskev : delo diplomskega seminarja

Petrovčič, Job (author), Strle, Sašo (mentor)

V uvodu je predstavljen pojem homotopije in izotopije. Delo obravnava zanke, fundamentalno grupo, geometrijsko število presečišč in dvokotniški kriterij. Definirana je grupa razredov preslikav. Ta grupa je določena za disk, enkrat preboden disk, kolobar, torus in enkrat preboden torus. Predstavljena je Alexandrova metoda in družina zank za Alexandrovo metodo za ploskve roda vsaj dva. V nadaljevanju so predstavljeni Dehnovi zasuki, temu sledi predstavitev njihovega delovanja na zanke. S pomočjo topoloških svežnjev je dokazano, da je Birmanovo zaporedje eksaktno. Del razprave je namenjen lastnostim Dehnovih zasukov. Na koncu je kot ključen prispevek dokazan Dehnov izrek, da je grupa razredov preslikav končno generirana z Dehnovimi zasuki.

• Language: Slovenian
• Keywords: geometrijska topologija, ploskev, homeomorfizem\sep difeomorfizem, homotopija, izotopija, grupa razredov preslikav, minimalna pozicija zank, fundamentalna grupa, Dehnov zasuk, Alexandrova metoda, končno generirana grupa, Birmanovo eksaktno zaporedje
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-150549
• UDC: 515.1
• COBISS.SI-ID: 165445379
• Publication date in RUL: 20.09.2023

Secondary language

• Language: English
• Title: Surface Diffeomorphisms

Abstract

First, terms homotopy and isotopy are discussed. Loops, fundamental group, geometric intersection number and bigon criterion are presented. Mapping class group is defined. This group is classified for disc, once-punctured disc, annulus, torus and once-punctured torus. Alexander method is presented and a family of loops for Alexander method is given for surfaces of genus at least two. Dehn twists are defined along with a presentation of their action on curves. Using fibre bundles the Birman sequence is shown to be exact. Basic properties of Dehn twists are proved. Finally, a proof of a theorem of Dehn that the mapping class group of a surface is finitely generated by Dehn twists is presented.

• Keywords: geometric topology, surface, homeomorphism, diffeomorphism, homotopy, isotopy, mapping class group, minimal position of loops, fundamental group, Dehn twist, Alexander method, finitely generated group, Birman exact sequence