Repairing the Universality Theorem for 4-Polytopes

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Perustieteiden korkeakoulu | Master's thesis
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Author
Date
2023-03-21
Department
Major/Subject
Mathematics
Mcode
SCI3054
Degree programme
Master’s Programme in Mathematics and Operations Research
Language
en
Pages
77
Series
Abstract
The realization space of a polytope P is defined as the space of all polytopes realizing the combinatorial type of P, modulo affine transformations. Richter-Gebert's universality theorem for 4-polytopes states that realization spaces of 4-polytopes can attain the homotopy type and algebraic complexity of any primary basic semialgebraic set. Richter-Gebert's proof relies on a new notion of stable equivalence between semialgebraic sets. The term has been used earlier in various forms and is meant to imply at least homotopy equivalence and equivalence of algebraic number type. However, due to a flaw in the definition, Richter-Gebert's version does not guarantee the former. In this thesis, the flaw in Richter-Gebert's definition is repaired through the introduction of a stricter, homotopy type preserving version of stable equivalence. It is then demonstrated that the rest of Richter-Gebert's original proof still works with the new definition. Lastly, the scattered history of stable equivalence is reviewed and reconciled.

Realisationsrymden av en polytop P är definierad som rymden av alla polytoper med samma kombinatoriella typ som P, modulo affina avbildningar. Richter-Geberts universalitetssats for 4-polytoper hävdar att realisationsrymden av en 4-polytop kan ha homotopitypen och algebraiska taltypen av en godtycklig primär semialgebraisk varietet. Richter-Geberts bevis är baserat på en ny version av stabil ekvivalens för semialgebraiska varieteter. Termen har tidigare använts flera gånger, och är menad att medföra både homotopiekvivalens och ekvivalens av algebraisk taltyp. Richter-Gebert's definition är dock felaktig i och med att den inte garanterar homotopiekvivalens. I denna avhandling korrigeras felet i Richter-Gebert's bevis med hjälp av en ny, starkare stabil ekvivalens. Det bevisas att den nya definitionen implicerar homotopiekvivalens, och därefter att resten av Richter-Gebert's bevis fortfarande fungerar. Avslutningsvis recenseras och förenas den existerande litteraturen angående stabil ekvivalens.
Description
Supervisor
Kubjas, Kaie
Thesis advisor
Boege, Tobias
Keywords
polytopes, discrete geometry, real algebraic geometry, topology
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https://urn.fi/URN:NBN:fi:aalto-202303262594
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[dipl] Perustieteiden korkeakoulu / SCI