Homotopska razdalja : delo diplomskega seminarja

Zaletelj, Alja (author), Pavešić, Petar (mentor)

V homotopski teoriji enačimo preslikave, ki so med seboj homotopne. Za poljubni preslikavi iz $X$ v $Y$ iščemo podprostore $X$, na katerih sta homotopni. Najmanjše število takih podprostorov, ki domeno $X$ pokrijejo, razglasimo za njuno homotopsko razdaljo. Z uporabo lastnosti homotopije in razširjanjem pokritij normalnih prostorov dokažemo, da je homotopska razdalja na njih metrika. Homotopsko razdaljo povežemo s Lusterik-Schnirelmannovo kategorijo in topološko kompleksnostjo. Povezave med njimi nam poenostavijo dokaze njihovih lastnosti in jih predstavijo v novi luči.

• Language: Slovenian
• Keywords: homotopija, homotopska razdalja, homotopska ekvivalenca, trikotniška neenakost, Lusternik-Schnirelmannova kategorija, kategorična množica, topološka kompleksnost, vlaknenje, prerezna kategorija
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-159199
• UDC: 515.1
• COBISS.SI-ID: 200520195
• Publication date in RUL: 03.07.2024

Secondary language

• Language: English
• Title: Homotopic distance

Abstract

In homotopy theory we identify maps that are homotopic. For two mappings from $X$ to $Y$ we look for subspaces of $X$ on which they are homotopic. The minimum number of such subspaces covering the domain $X$ is declared to be their homotopic distance. Using properties of homotopy and extending the covers of normal spaces, we prove that the homotopic distance on them is a metric. We connect homotopic distance with Lusternik-Schnirelmann category and topological complexity. The links between them simplify the proofs of their properties and present them in a new light.

• Keywords: homotopy, homotopic distance, homotopy equivalence, triangular inequality, Lusternik-Schnirelmann category, categorical set, topological complexity, fibration, sectional category