Browsing by Author "Hempel, John"

Now showing 1 - 10 of 10
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    Cone manifolds in three-dimensional topology applications to branched covers
    (1990) Jones, Kerry Nelson; Hempel, John
    Cone manifolds are defined and several standard geometric techniques for Riemannian manifolds are generalized to this setting. Smoothing techniques for approximating cone manifolds by smooth Riemannian manifolds with bounded sectional curvature are discussed. This involves some quite explicit curvature computations. The connection is then made between branched covers and cone manifolds by showing that cone manifold structures lift to a branched cover. Topological results concerning existence of incompressible tori and Seifert-fibered spaces in branched covers are then obtained by lifting cone manifold structures to a branched cover, smoothing the cone manifold structure to a bounded curvature metric, then using differential-geometric techniques on the smooth manifold. These results are then used in several explicit examples.
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    Construction of universal bundles
    (1979) Thongyoo, Sutep; Curtis, Morton L.; Hempel, John; Pfeiffer, Paul E.
    By a n-universal bundle, we mean a principal fiber bundle such that the bundle space is (n-1)-connected. The present paper shows that for a suitable base space X (namely an arc-wise connected space which can be covered by a countable number of contractible open sets) there exists an infinity-universal bundle over X.
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    Fully transitive polyhedra with crystallographic symmetry groups
    (1990) Eisenlohr, John Merrick; Hempel, John
    This dissertation sets forth a method for classifying, up to euclidean similarity, fully-transitive polyhedra which have crystallographic symmetry groups. Branko Grunbaum's definition of polyhedron is used, in which the faces are "hollow" - i.e., a face is a collection of edges. The approach is algorithmic: in a euclidean space $R\sp{n}$, assume we have classified the crystallographic groups up to conjugation by similarity transformations. For each similarity class of groups G, we follow a procedure which will generate all polyhedra in $R\sp{n}$ on which G acts fully-transitively. The method also assures that there is no duplication of examples. Following the presentation of this method, there is a discussion of the topology of these polyhedra. It is shown how to associate a non-compact surface with each polyhedron, and how to determine where this surface fits in the classification of non-compact 2-manifolds. Following this discussion, the method described is applied to the case of 2-dimensional crystallographic groups, and a complete classification of fully-transitive tilings is obtained.
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    High-distance splittings of 3-manifolds
    (2003) Marinenko, Tatiana; Hempel, John
    A Heegaard splitting (S; V1, V 2) for a closed 3-manifold M is a representation M = V1 ∪S V2 where V1 and V 2 are handlebodies and S = ∂V 1 = ∂V2 = V 1 ∩ V2. The distance of a Heegaard splitting (S; V1, V2) is the length of a shortest path in the curve complex of S which connects the subcomplexes KV1 and KV2 , where KVi is the subcomplex consisting of all vertices that correspond to simple closed curves bounding disks in Vi for i = 1, 2. In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that for any n the distance of the Heegaard Splitting of Mn is at least n. Moreover, we show that pi1(Mn) surjects onto pi1(Mn-1 ). Hence, if we assume that M0 has a non-trivial boundary, i.e. first Betti number beta1( M0) > 0, then it follows that beta1( Mn) > 0 for all n ≥ 1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n ≥ 1 and hyperbolizable 3-manifolds for n ≥ 3.
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    Homology of coverings of fibre bundles
    (1985) Chang, Shu-Chu; Hempel, John; Wolfson, Jon; Flapan, Erica
    We give a procedure for computing the rank of the first homology group of any finite sheeted covering space M of a fiber bundle M (over S1). Using the associated free abelian groups Fix g* and Fix g1, we find both upper and lower bounds for the rank of H1(M). In particular, using the free differential calculus and the Reidemeister-Schreier method, we derive a formula for computing rank H1(M) directly in a specific case.
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    Representations of low dimensional manifolds as branched coverings of spheres
    (1984) Austin, David M.; Hempel, John; Dunbar, William;Flapan, Erica
    We show that any 2- or 3-dimensional manifold is a branched covering of the sphere branched over a universal branching set. Using the associated unbranched covering, we show that there is a one-to-one correspondence between these branched coverings and pairs of permutations. In particular, this gives a means of studying manifolds. The goal of this work is to determine how much information about the manifold is readily accessible from the permutations.
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    Some conditions for recognizing a 3-manifold group
    (2010) Pershell, Karoline; Hempel, John
    In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a splitting is related to the fundamental group of a 3-manifold, still using diagrams as a tool to answer these questions. In the process, we determine an invariant of groups (by way of group presentations) for how far such presentations are from 3-manifolds.
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    Surface mapping classes and Heegaard decompositions of 3-manifolds
    (1990) Lu, Ning; Hempel, John
    The thesis is constituted in two parts. The first part including the first four chapters concentrates on the surface mapping class groups. The second part including the last three chapters focuses on their applications in the 3-manifold theory. Chapters I, II, and III show a new set of three generators, L, N and T of the surface mapping class groups ${\cal M}\sb{g}$ and investigate their topological and algebraic properties. Chapter IV finds a finite set of generators of the subgroup ${\cal K}\sb{g},$ which consists of the mapping classes that can be extended to a solid handlebody, in words of the generators of ${\cal M}\sb{g}$ given in the earlier chapters. Chapter V describes all Heegaard decompositions of the 3-sphere, and relates the homology 3-spheres to the elements of the Torelli subgroups. Chapter VI presents a new proof of the fundamental theorem of Kirby calculus on links by using the presentation of ${\cal M}\sb{g}$. The most important result of the thesis, which answers a question asked long ago about the stable equivalence of Heegaard decompositions of 3-manifolds, is proved in Chapter VII. We quote it here: Theorem VII.1.1. Any two Heegaard decompositions of the same genus of a 3-manifold of genus g are stably equivalent by adding no more than 3g - 3 trivial handles.
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    Using complexity bounds to study positive Heegaard diagrams of genus two
    (2001) Bellis, Amy Christner; Hempel, John
    The complexity of a Heegaard splitting is the minimal intersection number of two essential simple closed curves which bound disks on either side of the splitting. In order to study the complexity of a splitting, we discuss symmetries and other properties of positive genus two Heegaard diagrams. The complementary regions in such a diagram are either octagonal or square, and we are able to find upper and lower bounds on the complexity of the splitting represented by the diagram in terms of the number of complementary squares of each of nine types. We are then able to give obstructions to a manifold being Seifert fibered in terms of this data, in addition to showing that manifolds with diagrams of a particular type are Seifert fibered. We also discuss manifolds with a Heegaard splitting of complexity two or less, which are Seifert fibered. We show how to compute the orbit space and the Seifert invariants for these manifolds.
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    Using glueing diagrams to find boundary curves of incompressible surfaces in a hyperbolic knot space
    (1984) Cohn, Aaron I.; Veech, William A.; Culler, Marc; Shalen, Peter B.; Hempel, John
    We calculate some boundary curves of incompressible surfaces in some knot spaces. This method is based on a theorem of Marc Culler and Peter Shaien, and we make use of some calculations done by William Menasco.