Minimalni končni modeli topoloških prostorov : delo diplomskega seminarja

Bezjak, Filip (author), Pavešić, Petar (mentor)

Za končen topološki prostor $F$ rečemo, da je končen model prostora $X$, če obstaja šibka homotopska ekvivalenca $f: X \to F$. Model je minimalen, če ima izmed vseh modelov najmanjšo kardinalnost. Končnim $T_0$ topološkim prostorom lahko priredimo simplicialne komplekse. Eden izmed glavnih izrekov v delu nam pa pove, da je geometrijska realizacija tega kompleksa šibko homotopsko ekvivalentna začetnemu prostoru. S tem dobimo orodje za iskanje končnih modelov prostorov. V tem delu bomo poiskali minimalne modele topoloških sfer in končnih grafov.

• Language: Slovenian
• Keywords: končen topološki prostor, šibka homotopska ekvivalenca, homotopska ekvivalenca, delno urejena množica, simplicialni kompleks, graf, sfera
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-150818
• UDC: 515.1
• COBISS.SI-ID: 165560323
• Publication date in RUL: 24.09.2023

Secondary language

• Language: English
• Title: Minimal finite models

Abstract

We say that a finite topological space $F$ is a model of a topological space $X$ if there exists a weak homotopy equivalence $f: X \to F$. A model is minimal if it has the smallest cardinality of all models of a space. Every finite $T_0$ space has its associated simplicial complex. One of the main theorems of this graduate thesis states that the geometric realization of the associated simplicial complex is weak homotopy equivalent to the initial space. This gives us a tool to find finite models of spaces. In this thesis, we will find minimal models of topological spheres and finite graphs.

• Keywords: finite topological space, weak homotopy equivalence, homotopy equivalence, partialy ordered set, simplicial complex, graph, sphere