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Title: Sphere systems in 3-manifolds and arc graphs
Author: Iezzi, Francesca
ISNI:       0000 0004 5924 2098
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2016
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We present in this thesis some results about sphere graphs of 3-manifolds. If we denote as Mg the connected sum of g copies of S2 x S1, the sphere graph of Mg, denoted as S(Mg), is the graph whose vertices are isotopy classes of essential spheres in Mg, where two vertices are adjacent if the spheres they represent can be realised disjointly. Sphere graphs have turned out to be an important tool in the study of outer automorphisms groups of free groups. The thesis is mainly focused on two projects. As a first project, we develop a tool in the study of sphere graphs, via analysing the intersections of two collections of spheres in the 3-manifold Mg. Elaborating on Hatcher's work and on his definition of normal form for spheres ([15]), we define a standard form for two embedded sphere systems (i.e collections of disjoint spheres) in Mg. We show that such a standard form exists for any couple of maximal sphere systems in Mg, and is unique up to homeomorphisms of Mg inducing the identity on the fundamental group. Our proof uses combinatorial and topological methods. We basically show that most of the information about two embedded maximal sphere systems in Mg is contained in a 2-dimensional CW complex, which we call the square complex associated to the two sphere systems. The second project concerns the connections between arc graphs of surfaces and sphere graphs of 3-manifolds. If S is a compact orientable surface whose fundamental group is the free group Fg, then there is a natural injective map i from the arc graph of the surface S to the sphere graph of the 3-manifold Mg. It has been proved ([12]) that this map is an isometric embedding. We prove, using topological methods, that the map i admits a coarsely defined Lipschitz left inverse.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics