The triangulation of manifolds

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.Comment: 26 pages, 2 figures. version 2: spellings corrected, analytic speculations in 4.8.2 sharpene

The Configuration Space of Two Particles Moving on a Graph

In this thesis we study the configuration space, F (Γ, 2), of two particles moving without collisions on a graph Γ with a view to calculating the Betti numbers of this space. We develop an intersection theory for cycles in graphs inspired by the classical intersection theory for cycles in manifolds and we use this to develop an algorithm to calculate the second Betti number of F (Γ,2) for any graph Γ. We also use this intersection theory to provide a complete description of the cohomology algebra H ^*(F (Γ, 2), Q) for any planar graph Γ and to calculate explicit formulae for the Betti numbers of F (Γ, 2) when Γ is a complete graph or a complete bipartite graph. We also investigate the generators of group H_2 (F (Γ, 2), Z) and show that for any planar graph this group is entirely generated by tori induced by disjoint cycles in the graph. For non-planar graphs the situation is more complicated and we show that there can exist a generator of H_2 (F (Γ, 2), Z) which is not the fundamental class of a surface embedded in the space F (Γ, 2)

Involutions sur les variétés de dimension trois et homologie de Khovanov

Cette thèse établit, et étudie, un lien entre l'homologie de Khovanov et la topologie des revêtements ramifiés doubles. Nous y introduisons certaines propriétés de stabilité en homologie de Khovanov, dont nous dérivons par la suite des obstructions à l'existence de certaines chirurgies exceptionnelles sur les noeuds admettant une involution\ud appropriée. Ce comportement, analogue à celui de l'homologie de Heegaard-Floer sous chirurgie, renforce ainsi le lien existant (dû à Ozsváth et Szabó) entre homologie de Khovanov, et homologie d'Heegaard-Floer des revêtements ramifiés doubles. Dans l'optique de poursuivre et d'exploiter plus avant cette relation, les méthodes développées dans ce travail sont appliquées à l'étude des L-espaces, et à déterminer, en premier lieu, si l'homologie de Khovanov fournit un invariant des revêtements ramifiés doubles, et en deuxième lieu, si l'homologie de Khovanov permet de détecter le noeud trivial. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Homologie de Khovanov, Homologie de Heegaard-Floer, Chirurgies de Dehn, Involutions, Variétés de dimension trois, Revêtements ramifiés doubles

Betti Cones of Stanley-Reisner Ideals

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. Betti diagrams encode information about the minimal free graded resolutions of these ideals, and are therefore important algebraic invariants. Computing resolutions is a difficult task in general, but in our case there are tools we can use to simplify it. Most immediately, the Stanley-Reisner Correspondence assigns a unique simplicial complex to every squarefree monomial ideal, and Hochster’s Formula allows us to compute the Betti diagrams of these ideals from combinatorial properties of their corresponding complexes. This reduces the algebraic problem of computing resolutions to the (often easier) combinatorial problem of computing homologies. As such, most of our work is combinatorial in nature. The other key tool we use in studying these diagrams is Boij-Soderberg Theory. This theory views Betti diagrams as vectors in a rational vector space, and investigates them by considering the convex cone they generate. This technique has proven very instructive: it has allowed us to classify all Betti diagrams up to integer multiplication. This thesis applies the theory more narrowly, to the cones generated by diagrams of squarefree monomial ideals. In Chapter 2 we introduce all of these concepts, along with some preliminary results in both algebra and combinatorics we will need going forward. Chapter 3 presents the dimensions of our cones, along with the vector spaces they span. Chapters 4 and 5 are devoted to the pure Betti diagrams in these cones, and the combinatorial properties of their associated complexes. Finally Chapter 6 builds on these results to prove a partial analogue of the first Boij-Soderberg conjecture for squarefree monomial ideals, by detailing an algorithm for generating pure Betti diagrams of squarefree monomial ideals of any degree type

Über endlich erzeugte Gruppen und bestimmte Funktoren

We consider chains of finitely presentable groups with additional properties. These additional properties ensure that every finitely generated group can be written as the colimit of such a chain. We prove various results about these chains, including an isomorphism theorem, and show that the category of these chains can be used to describe the category of finitely generated groups "up to natural isomorphism". The last part is an instance of localisations of categories, and is formulated as an equivalence of categories. Finally, we consider the homology of finitely generated groups. We apply spectral sequences and equivariant homology groups to derive long, exact sequences for the homology of three special classes of finitely generated groups.Wir betrachten Ketten von endlich präsentierbaren Gruppen mit bestimmten Einschränkungen. Diese Einschränkungen garantieren, dass jede endlich erzeugte Gruppe als Kolimit einer derartigen Kette aufgefasst werden kann. Für diese Ketten beweisen wir verschiedene Resultate, insbesondere einen Isomorphiesatz. Als wichtigstes Resultat dieser Arbeit zeigen wir, dass die Kategorie dieser Ketten verwendet werden kann, um die Kategorie der endlich erzeugten Gruppen zu beschreiben. Dies wird formuliert als Äquivalenz von Kategorien. Schließlich leiten wir für drei spezielle Klassen von endlich erzeugten Gruppen lange, exakte Homologiesequenzen her, indem wir äquivariante Homologie und Spektralsequenzen verwenden

PSA 2016

These preprints were automatically compiled into a PDF from the collection of papers deposited in PhilSci-Archive in conjunction with the PSA 2016

PSA 2016

These preprints were automatically compiled into a PDF from the collection of papers deposited in PhilSci-Archive in conjunction with the PSA 2016

PSA 2016

These preprints were automatically compiled into a PDF from the collection of papers deposited in PhilSci-Archive in conjunction with the PSA 2016