arXiv subjects

Ulrich Oertel

Publications and source records attributed to Ulrich Oertel.

11 recordsLinked to original sources

arXiv2018

Lamination links in 3-manifolds

We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links bound 2-dimensional measured and oriented "Seifert laminations." We show that any lamination link which bounds a 2-dimensional Seifert lamination, bounds a "taut" Seifert lamination, i.e. one of maximum Euler characteristic, subject to the condition that the Seifert lamination is carried by an aspherical branched surface. This maximum Euler characteristic function is continuous on certain parametrized families of lamination links carried by a train track neighborhood. Taut Seifert laminations generalize minimal genus Seifert surfaces.

math.GT↗

arXiv2018

A Seifert algorithm for lamination links

We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and are carried by an $S^1$-fibered tube neighborhood of an oriented train track embedded in a 3-manifold.

math.GT↗

arXiv2016

Laminations with transverse measures in ordered abelian semigroups

We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure $\mathbb L$ and measurable space $X$ we define $\mathbb L$-measures $ν$ on $X$. If $L$ is a codimension-1 lamination in a manifold, it often admits transverse $\mathbb L$-measures for some $\mathbb L$. Transverse $\mathbb L$-measures can be used to understand classes of laminations much larger than the class of laminations admitting transverse positive $\mathbb R$-measures. In particular, we show that "finite or infinite depth measured laminations" are laminations admitting transverse measures with values in a certain ordered semiring $\bar{\mathbb O}$ satisfying the additional property that locally the values lie in a smaller semiring $\mathbb P$. We consider the "realization problem:" In one version, this deals with the problem whether an $\mathbb P$-invariant weight vector assigned to a branched manifold $B$ (satisfying certain branch equations) determines a lamination $L$ carried by $B$ with a transverse $\bar{\mathbb O}$-measure inducing the weights on $B$. We describe further laminations which may not be $\mathbb L$-measured, but are "well-covered" by laminations with transverse $\mathbb L$-measures. We also investigate actions on $\mathbb L$-trees which are associated to essential laminations with transverse $\mathbb L$-measures. In appendices, we develop ideas about $\mathbb L$-measures a little further, for example showing that a $\mathbb P$-measure can be interpreted as a kind of probability measure.

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arXiv2014

Measured lamination spaces for surface pairs

We calculate a projective space of essential measured laminations in a surface pair, which will be used in another paper to help describe spaces of "finite height laminations."

math.GT↗

arXiv2014

Finite height lamination spaces for surfaces

We describe spaces of essential finite height (measured) laminations in a surface $S$ using a parameter space we call $\mathbb S$, an ordered semi-ring. We show that for every finite height essential lamination $L$ in $S$, there is an action of $π_1(S)$ on an $\mathbb S$-tree dual to the lift of $L$ to the universal cover of $S$.

math.GT↗

arXiv2010

Essential disks and semi-essential surfaces in 3-manifolds

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal surface theory will be used in other papers to describe objects representing limits of essential disks in a handlebody or a 3-manifold with compressible boundary. For certain automorphisms of handlebodies, these disk limits should serve as invariant objects akin to laminations and analogous to the invariant laminations for pseudo-Anosov automorphisms of surfaces.

math.GT↗

arXiv2007

Mapping class groups of compression bodies and 3-manifolds

We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping class group of discrepant automorphisms of W in Q. We describe a short exact sequence of these mapping class groups. For an orientable, compact, reducible 3 manifold W, there is a canonical "maximal" 4-dimensional compression body Q whose exterior boundary is W and whose interior boundary is the disjoint union of the irreducible summands of W. We obtain a short exact sequence for the mapping class group of a 3-manifold, which gives the mapping class group of the disjoint union of irreducible summands as a quotient of the entire mapping class group by the group of adjusting automorphisms. The group of discrepant automorphisms is described in terms of generators. The results are useful in a program for classifying automorphisms of compact 3-manifolds in the spirit of Nielsen-Thurston.

math.GT↗

arXiv2005

A classification of automorphisms of compact 3-manifolds

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as a suitable composition of two automorphisms, each of which fits into our classification. Given an automorphism, the goal is to show, loosely speaking, either that it is periodic, or that it can be decomposed on a surface invariant up to isotopy, or that it has a "dynamically nice" representative, with invariant laminations that "fill" the manifold. We consider automorphisms of irreducible and boundary-irreducible 3-manifolds as being already classified, though there are some exceptional manifolds for which the automorphisms are not understood. Thus the paper is particularly aimed at understanding automorphisms of reducible and/or boundary reducible 3-manifolds. Previously unknown phenomena are found even in the case of connected sums of products of a 2-sphere with a 1-sphere. To deal with this case, we prove that a minimal genus Heegaard decomposition is unique up to isotopy, a result which apparently was previously unknown. Much remains to be understood about some of the automorphisms of the classification.

math.GT↗

arXiv2003

Contact Structures, sigma-confoliations, and contaminations in 3-manifolds

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called sigma-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the sigma-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures.

math.GT↗

arXiv2003

A contamination carrying criterion for branched surfaces

A contamination in a 3-manifold is an object interpolating between the contact structure and the lamination. Contaminations seem to provide a link between 3-dimensional contact geometry and the classical topology of 3-manifolds, as described in a separate paper. In this paper we deal with contaminations carried by branched surfaces, giving a sufficient condition for a branched surface to carry a pure contamination.

math.GT↗

arXiv2003

Incompressible maps of surfaces and Dehn filling

We prove results showing that the existence of essential maps of surfaces in a manifold M' obtained from a 3-manifold M by Dehn filling implies the existence of essential maps of surfaces in M.

math.GT↗