Courses:
      Basic Calculus   21:640:119:11; Tuesday, Thursday 2:30-3:50,  Conklin Hall, Room 453
      Applied Mathematical Analysis I,  21:640:475:01; Tuesday, Thursday 11:30-12:50 pm, Smith Hall, Room 234

      Course web pages on Blackboard

Office Hours:  Tuesday 4:30-5:30 pm, Thursday 10:20-11:20 am, and by appointment;

Phone - 973-353-5156 Ext 3909;           Email - oertel@andromeda.rutgers.edu

Department of Mathematics and Computer Science, Rutgers-Newark

Current Research 

Automorphisms of 3-manifolds

The most interesting automorphisms (self-homeomorphisms) of 3-manifolds occur in boundary-reducible and reducible manifolds. The paper Automorphisms of 3-dimensional handlebodies develops a theory of automorphisms of handlebodies and compression bodies analogous to the Nielsen-Thurston theory of autmorphisms of surfaces. In the theory, the analogue of the pseudo-Anosov automorphism is the generic automorphism of a handlebody or compression body, and generic automorphisms turn out, as one would expect, to be the most interesting and mysterious.   Automorphisms of irreducible 3-manifolds with non-empty boundary can then be understood by combining the new methods with older methods in 3-manifold topology, involving the Jaco-Shalen-Johannson characteristic submanifold and Bonahon's characteristic compression body.  Much remains to be understood about generic automorphisms, which are currently being studied further by former Rutgers graduate student Leonardo Navarro de Carvalho.

In recent work with Carvalho, the ideas used for the classification of automorphisms of handlebodies and compression bodies are extended to classify, in a certain sense, the automorphisms of any compact 3-manifold (satisfying the Thurston Geometrization Conjecture).

Automorphisms of 3-dimensional handlebodies and compression bodies, Topology 41 (2002) 363-410:      Postscript file       PDF file

A classification of automorphisms of compact 3-manifolds,  with Leonardo Navarro de Carvalho.   Math arXiv
 

Mapping class groups of compression bodies and 3-manifolds

In work related to the classification of automorphisms of 3-manifolds, I obtain a short exact sequence relating various mapping class groups (diffeotopy groups) associated to a compression body of arbitrary dimension.  This is applied to obtain a short exact sequence for the mapping class group of a reducible 3-manifold M.  The sequence gives the mapping class group of the disjoint union of irreducible summands of M as a quotient of the entire mapping class group of M.  There are further applications.

Mapping class groups of compression bodies and 3-manifolds.Math arXiv

Contact structures and contaminations in 3-manifolds

In a project with Jacek Swiatkowski (Wroclaw University) I am studying contact structures using branched surface techniques. A natural generalization of the contact structure and the confoliation is the contamination, an object which can be carried by a branched surface. Contact structures can be represented by contaminations. The goals of this research are 1) to understand contact structures combinatorially; 2) to understand the relationships between foliations, confoliations, contact structures, laminations, and contaminations; and 3) ultimately to apply the methods to obtain information about 3-manifold topology.

A contamination carrying criterion for branched surfaces, Preprint, 2001.      Math arXiv

Contact structures, sigma-confoliations, and contaminations in 3-manifolds, Preprint, 2002.      Math arXiv

Maps of surfaces into 3-manifolds

Embedded incompressible surfaces are at the heart of classical 3-manifold topology. Among possible generalizations are essential maps of surfaces into 3-manifolds. Usually a map is called essential if it induces an injection on the fundamental group. If the Simple Loop Conjecture for maps from surfaces to 3-manifolds is true, then this is equivalent to a weaker notion of essentiality which I have studied in the paper Incompressible maps of surfaces: boundary slopes and Dehn filling . Goals of this research include studying these essential surfaces in greater detail and proving the Simple Loop Conjecture. Another related paper is Boundaries of pi₁ -injective surfaces, see below.

Incompressible maps of surfaces: boundary slopes and Dehn filling, Preprint, 1998:    Math arXiv

Some Papers

• On the existence of infinitely many essential surfaces of bounded genus, Pac. J. Math.  202, No. 2, 2002, 449-458. Postscript file
• Spaces which are not negatively curved, joint with Lee Mosher. Communications in Analysis and Geometry, 6 , No1, (1998), 67-140.
• Two-dimensional measured laminations of positive Euler characteristic, joint with Lee Mosher.  Oxford Q. J. Math. 52 (2001), no. 2, 195--216.
• Invariant laminations for diffeomorphisms of handlebodies, Oxford Quarterly Journal of Mathematics, 51, No. 4, (2000)
• 485-507.
• Boundaries of pi₁-injective surfaces, Topology and its Applications 78 (1997), 215-234.
• Affine foliations and covering hyperbolic structures, joint with Athanase Papadopoulos, Manuscripta Mathematica 104 (2001) 383-406.
• Affine laminations and their stretch factors, Pacific Journal of Mathematics 182, No. 2, (1998), 303-327.
• Full laminations in 3-manifolds, joint with Allen Hatcher, Math. Proc. Camb. Phil. Soc. 119 (1996), 73-82.
• Affine lamination spaces for surfaces, joint with Allen Hatcher, Pac. J. of Math. 154 (1992), No.1, 87-101.
• Boundary slopes for Montesinos knots, joint with Allen Hatcher, Topology 28 (1989), 453-480.
• Essential laminations in 3-manifolds, joint with David Gabai, Annals of Mathematics 130 (1989), 41-73.
• Sums of incompressible surfaces, Proceedings AMS 102 (1988) No.3, 711-719.
• Measured laminations in 3-manifolds, Transactions AMS. 305 (1988) No. 2, 531-573.
• Homology branched surfaces: Thurston's norm on H₂(M ³), LMS Lecture Note Series 112, Low-dimensional Topology and Kleinian Groups, Editor D.B.A. Epstein, 1986.
• An algorithm to decide if a 3-manifold is a Haken manifold, joint with William Jaco, Topology 23 (1984), 195-209 .
• Incompressible branched surfaces, Invent. Math. 76 (1984), 385-410.
• Closed incompressible surfaces in complements of star links, Pac. J. of Math. 111 (1984), 209-230.
• Incompressible surfaces via branched surfaces, joint with William Floyd, Topology 23 (1984), 117-125.